Displaying 1-10 of 40 results found.
Rank array R of 3/2 read by antidiagonals; this array is the dispersion of the complement of the sequence given by r(n) = r(n-1) + 1 + floor(3n/2) for n>=1, with r(0) = 1; that is, A077043(n+1).
+20
7
1, 2, 3, 4, 5, 7, 6, 8, 10, 12, 9, 11, 14, 16, 19, 13, 15, 18, 21, 24, 27, 17, 20, 23, 26, 30, 33, 37, 22, 25, 29, 32, 36, 40, 44, 48, 28, 31, 35, 39, 43, 47, 52, 56, 61, 34, 38, 42, 46, 51, 55, 60, 65, 70, 75, 41, 45, 50, 54, 59, 64, 69, 74, 80, 85, 91, 49, 53, 58, 63, 68, 73
COMMENTS
The sequence is a permutation of the positive integers and the array is a transposable dispersion.
Let T(n,k) be the rectangular version of the array at A036561, with northwest corner as shown here: 1 2 4 8 16 32
3 6 12 24 48 96
9 18 36 72 144 288
27 54 108 216 432 864
Then R(n,k) is the rank of T(n,k) when all the numbers in {T(n,k)} are jointly ranked. - Clark Kimberling, Jan 25 2018
FORMULA
R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.
EXAMPLE
Northwest corner of R:
1 2 4 6 9 13 17 22
3 5 8 11 15 20 25 31
7 10 14 18 23 29 35 42
12 16 21 26 32 39 46 54
19 24 30 36 43 51 59 68
27 33 40 47 55 64 73 83
37 44 52 60 69 79 89 100
Let t=3/2; then R(i,j) = rank of (j,i) when all nonnegative integer pairs (a,b) are ranked by the relation << defined as follows: (a,b) << (c,d) if a + b*t < c + d*t, and also (a,b) << (c,d) if a + b*t = c + d*t and b < d. Thus R(2,1) = 10 is the rank of (1,2) in the list (0,0) << (1,0) << (0,1) << (2,0) << (1,1) << (3,0) << (0,2) << (2,1) << (4,0) << (1,2).
MATHEMATICA
r = 20; r1 = 12; (*r=# rows of T, r1=# rows to show*);
c = 20; c1 = 12; (*c=# cols of T, c1=# cols to show*);
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[3 n/2]; u = Table[s[n], {n, 0, 100}]
v = Complement[Range[Max[u]], u]; f[n_] := v[[n]]; Table[f[n], {n, 1, 30}]
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]; rows = {NestList[f, 1, c]}; Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}]; w[i_, j_] := rows[[i, j]];
TableForm[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A087465 array *) Flatten[Table[w[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A087465 sequence *) TableForm[Table[w[i, 1] + w[1, j] + (i - 1)*(j - 1) - 1, {i, 1, 10}, {j, 1, 10}]] (* A087465 array, by formula *)
Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4).
(Formerly M0998 N0374)
+10
499
0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, 132, 144, 156, 169, 182, 196, 210, 225, 240, 256, 272, 289, 306, 324, 342, 361, 380, 400, 420, 441, 462, 484, 506, 529, 552, 576, 600, 625, 650, 676, 702, 729, 756, 784, 812
COMMENTS
b(n) = a(n+2) is the number of multigraphs with loops on 2 nodes with n edges [so g.f. for b(n) is 1/((1-x)^2*(1-x^2))]. Also number of 2-covers of an n-set; also number of 2 X n binary matrices with no zero columns up to row and column permutation. - Vladeta Jovovic, Jun 08 2000 a(n) is also the maximal number of edges that a triangle-free graph of n vertices can have. For n = 2m, the maximum is achieved by the bipartite graph K(m, m); for n = 2m + 1, the maximum is achieved by the bipartite graph K(m, m + 1). - Avi Peretz (njk(AT)netvision.net.il), Mar 18 2001
a(n) is the number of arithmetic progressions of 3 terms and any mean which can be extracted from the set of the first n natural numbers (starting from 1). - Santi Spadaro, Jul 13 2001 This is also the order dimension of the (strong) Bruhat order on the Coxeter group A_{n-1} (the symmetric group S_n). - Nathan Reading (reading(AT)math.umn.edu), Mar 07 2002
Let M_n denote the n X n matrix m(i,j) = 2 if i = j; m(i, j) = 1 if (i+j) is even; m(i, j) = 0 if i + j is odd, then a(n+2) = det M_n. - Benoit Cloitre, Jun 19 2002 Sums of pairs of neighboring terms are triangular numbers in increasing order. - Amarnath Murthy, Aug 19 2002 Also, from the starting position in standard chess, minimum number of captures by pawns of the same color to place n of them on the same file (column). Beyond a(6), the board and number of pieces available for capture are assumed to be extended enough to accomplish this task. - Rick L. Shepherd, Sep 17 2002 For example, a(2) = 1 and one capture can produce "doubled pawns", a(3) = 2 and two captures is sufficient to produce tripled pawns, etc. (Of course other, uncounted, non-capturing pawn moves are also necessary from the starting position in order to put three or more pawns on a given file.) - Rick L. Shepherd, Sep 17 2002 Terms are the geometric mean and arithmetic mean of their neighbors alternately. - Amarnath Murthy, Oct 17 2002 a(n+1) gives number of non-symmetric partitions of n into at most 3 parts, with zeros used as padding. E.g., a(6) = 12 because we can write 5 = 5 + 0 + 0 = 0 + 5 + 0 = 4 + 1 + 0 = 1 + 4 + 0 = 1 + 0 + 4 = 3 + 2 + 0 = 2 + 3 + 0 = 2 + 0 + 3 = 2 + 2 + 1 = 2 + 1 + 2 = 3 + 1 + 1 = 1 + 3 + 1. - Jon Perry, Jul 08 2003 a(n-1) gives number of distinct elements greater than 1 of non-symmetric partitions of n into at most 3 parts, with zeros used as padding, appear in the middle. E.g., 5 = 5 + 0 + 0 = 0 + 5 + 0 = 4 + 1 + 0 = 1 + 4 + 0 = 1 + 0 + 4 = 3 + 2 + 0 = 2 + 3 + 0 = 2 + 0 + 3 = 2 + 2 + 1 = 2 + 1 + 2 = 3 + 1 + 1 = 1 + 3 + 1. Of these, 050, 140, 320, 230, 221, 131 qualify and a(4) = 6. - Jon Perry, Jul 08 2003 Conjectured size of the smallest critical set in a Latin square of order n (true for n <= 8). - Richard Bean, Jun 12 2003 and Nov 18 2003 a(n) gives number of maximal strokes on complete graph K_n, when edges on K_n can be assigned directions in any way. A "stroke" is a locally maximal directed path on a directed graph. Examples: n = 3, two strokes can exist, "x -> y -> z" and " x -> z", so a(3) = 2. n = 4, four maximal strokes exist, "u -> x -> z" and "u -> y" and "u -> z" and "x -> y -> z", so a(4) = 4. - Yasutoshi Kohmoto, Dec 20 2003 Number of symmetric Dyck paths of semilength n+1 and having three peaks. E.g., a(4) = 4 because we have U*DUUU*DDDU*D, UU*DUU*DDU*DD, UU*DDU*DUU*DD and UUU*DU*DU*DDD, where U = (1, 1), D = (1, -1) and * indicates a peak. - Emeric Deutsch, Jan 12 2004 Number of valid inequalities of the form j + k < n + 1, where j and k are positive integers, j <= k, n >= 0. - Rick L. Shepherd, Feb 27 2004 See A092186 for another application. Also, the number of nonisomorphic transversal combinatorial geometries of rank 2. - Alexandr S. Radionov (rasmailru(AT)mail.ru), Jun 02 2004
a(n+1) is the transform of n under the Riordan array (1/(1-x^2), x). - Paul Barry, Apr 16 2005 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ... specifies the largest number of copies of any of the gifts you receive on the n-th day in the "Twelve Days of Christmas" song. For example, on the fifth day of Christmas, you have 9 French hens. - Alonso del Arte, Jun 17 2005 a(n+1) is the number of noncongruent integer-sided triangles with largest side n. - David W. Wilson [Comment corrected Sep 26 2006] A quarter-square table can be used to multiply integers since n*m = a(n+m) - a(n-m) for all integer n, m. - Michael Somos, Oct 29 2006 The sequence is the size of the smallest strong critical set in a Latin square of order n. - G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), Feb 16 2007
Maximal number of squares (maximal area) in a polyomino with perimeter 2n. - Tanya Khovanova, Jul 04 2007 For n >= 3 a(n-1) is the number of bracelets with n+3 beads, 2 of which are red, 1 of which is blue. - Washington Bomfim, Jul 26 2008 Also a(n) is the number of different patterns of a 2-colored 3-partition of n. - Ctibor O. Zizka, Nov 19 2014 Also a(n-1) = C(((n+(n mod 2))/2), 2) + C(((n-(n mod 2))/2), 2), so this is the second diagonal of A061857 and A061866, and each even-indexed term is the average of its two neighbors. - Antti Karttunena(n) gives the number of nonisomorphic faithful representations of the Symmetric group S_3 of dimension n. Any faithful representation of S_3 must contain at least one copy of the 2-dimensional irrep, along with any combination of the two 1-dimensional irreps. - Andrew Rupinski, Jan 20 2011 a(n+2) gives the number of ways to make change for "c" cents, letting n = floor(c/5) to account for the 5-repetitive nature of the task, using only pennies, nickels and dimes (see A187243). - Adam Sasson, Mar 07 2011 a(n) belongs to the sequence if and only if a(n) = floor(sqrt(a(n))) * ceiling(sqrt(a(n))), that is, a(n) = k^2 or a(n) = k*(k+1), k >= 0. - Daniel Forgues, Apr 17 2011 a(n) is the sum of the positive integers < n that have the opposite parity as n.
Deleting the first 0 from the sequence results in a sequence b = 0, 1, 2, 4, ... such that b(n) is sum of the positive integers <= n that have the same parity as n. The sequence b(n) is the additive counterpart of the double factorial. - Peter Luschny, Jul 06 2011 Written as a(1) = 1, a(n) = a(n-1) + ceiling (a(n-1)) this is to ceiling as A002984 is to floor, and as A033638 is to round. - Jonathan Vos Post, Oct 08 2011 a(n-2) gives the number of distinct graphs with n vertices and n regions. - Erik Hasse, Oct 18 2011 Construct the n-th row of Pascal's triangle ( A007318) from the preceding row, starting with row 0 = 1. a(n) counts the total number of additions required to compute the triangle in this way up to row n, with the restrictions that copying a term does not count as an addition, and that all additions not required by the symmetry of Pascal's triangle are replaced by copying terms. - Douglas Latimer, Mar 05 2012 a(n) is the sum of the positive differences of the parts in the partitions of n+1 into exactly 2 parts. - Wesley Ivan Hurt, Jan 27 2013 a(n) is the maximum number of covering relations possible in an n-element graded poset. For n = 2m, this bound is achieved for the poset with two sets of m elements, with each point in the "upper" set covering each point in the "lower" set. For n = 2m+1, this bound is achieved by the poset with m nodes in an upper set covering each of m+1 nodes in a lower set. - Ben Branman, Mar 26 2013 a(n+2) is the number of (integer) partitions of n into 2 sorts of 1's and 1 sort of 2's. - Joerg Arndt, May 17 2013 Alternative statement of Oppermann's conjecture: For n>2, there is at least one prime between a(n) and a(n+1). - Ivan N. Ianakiev, May 23 2013. [This conjecture was mentioned in A220492, A222030. - Omar E. Pol, Oct 25 2013] For any given prime number, p, there are an infinite number of a(n) divisible by p, with those a(n) occurring in evenly spaced clusters of three as a(n), a(n+1), a(n+2) for a given p. The divisibility of all a(n) by p and the result are given by the following equations, where m >= 1 is the cluster number for that p: a(2m*p)/p = p*m^2 - m; a(2m*p + 1)/p = p*m^2; a(2m*p + 2)/p = p*m^2 + m. The number of a(n) instances between clusters is 2*p - 3. - Richard R. Forberg, Jun 09 2013 Apart from the initial term this is the elliptic troublemaker sequence R_n(1,2) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences R_n(a,b) see the cross references below. - Peter Bala, Aug 08 2013 a(n) is also the total number of twin hearts patterns (6c4c) packing into (n+1) X (n+1) coins, the coins left is A042948 and the voids left is A000982. See illustration in links. - Kival Ngaokrajang, Oct 24 2013 Partitions of 2n into parts of size 1, 2 or 4 where the largest part is 4, i.e., A073463(n,2). - Henry Bottomley, Oct 28 2013 a(n+1) is the minimum length of a sequence (of not necessarily distinct terms) that guarantees the existence of a (not necessarily consecutive) subsequence of length n in which like terms appear consecutively. This is also the minimum cardinality of an ordered set S that ensures that, given any partition of S, there will be a subset T of S so that the induced subpartition on T avoids the pattern ac/b, where a < b < c. - Eric Gottlieb, Mar 05 2014 Also the number of elements of the list 1..n+1 such that for any two elements {x,y} the integer (x+y)/2 lies in the range ]x,y[. - Robert G. Wilson v, May 22 2014 Number of lattice points (x,y) inside the region of the coordinate plane bounded by x <= n, 0 < y <= x/2. For a(11)=30 there are exactly 30 lattice points in the region below:
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0 1 2 3 4 5 6 7 8 9 10 11 .. n
a(n+1) is the greatest integer k for which there exists an n x n matrix M of nonnegative integers with every row and column summing to k, such that there do not exist n entries of M, all greater than 1, and no two of these entries in the same row or column. - Richard Stanley, Nov 19 2014 In a tiling of the triangular shape T_N with row length k for row k = 1, 2, ..., N >= 1 (or, alternatively row length N = 1-k for row k) with rectangular tiles, there can appear rectangles (i, j), N >= i >= j >= 1, of a(N+1) types (and their transposed shapes obtained by interchanging i and j). See the Feb 27 2004 comment above from Rick L. Shepherd. The motivation to look into this came from a proposal of Kival Ngaokrajang in A247139. - Wolfdieter Lang, Dec 09 2014 Every positive integer is a sum of at most four distinct quarter-squares; see A257018. - Clark Kimberling, Apr 15 2015 a(n+1) gives the maximal number of distinct elements of an n X n matrix which is symmetric (w.r.t. the main diagonal) and symmetric w.r.t. the main antidiagonal. Such matrices are called bisymmetric. See the Wikipedia link. - Wolfdieter Lang, Jul 07 2015 a(n) is the number of partitions of 2n+1 of length three with exactly two even entries (see below example). - John M. Campbell, Jan 29 2016 a(n) is the sum of the asymmetry degrees of all 01-avoiding binary words of length n. The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. a(6) = 9 because the 01-avoiding binary words of length 6 are 000000, 100000, 110000, 111000, 111100, 111110, and 111111, and the sum of their asymmetry degrees is 0 + 1 + 2 + 3 + 2 + 1 + 0 = 9. Equivalently, a(n) = Sum_{k>=0} k* A275437(n,k). - Emeric Deutsch, Aug 15 2016 a(n) is the number of ways to represent all the integers in the interval [3,n+1] as the sum of two distinct natural numbers. E.g., a(7)=12 as there are 12 different ways to represent all the numbers in the interval [3,8] as the sum of two distinct parts: 1+2=3, 1+3=4, 1+4=5, 1+5=6, 1+6=7, 1+7=8, 2+3=5, 2+4=6, 2+5=7, 2+6=8, 3+4=7, 3+5=8. - Anton Zakharov, Aug 24 2016 a(n+2) is the number of conjugacy classes of involutions (considering the identity as an involution) in the hyperoctahedral group C_2 wreath S_n. - Mark Wildon, Apr 22 2017 a(n+2) is the maximum number of pieces of a pizza that can be made with n cuts that are parallel or perpendicular to each other. - Anton Zakharov, May 11 2017 Also the matching number of the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017 The answer to a question posed by W. Mantel: a(n) is the maximum number of edges in an n-vertex triangle-free graph. Also solved by H. Gouwentak, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff. - Charles R Greathouse IV, Feb 01 2018 Maximum area of a rectangle with perimeter 2n and sides of integer length. - André Engels, Jul 29 2018 Also the crossing number of the complete bipartite graph K_{3,n+1}. - Eric W. Weisstein, Sep 11 2018 a(n+2) is the number of distinct genotype frequency vectors possible for a sample of n diploid individuals at a biallelic genetic locus with a specified major allele. Such vectors are the lists of nonnegative genotype frequencies (n_AA, n_AB, n_BB) with n_AA + n_AB + n_BB = n and n_AA >= n_BB. - Noah A Rosenberg, Feb 05 2019 a(n+2) is the number of distinct real spectra (eigenvalues repeated according to their multiplicity) for an orthogonal n X n matrix. The case of an empty spectrum list is logically counted as one of those possibilities, when it exists. Thus a(n+2) is the number of distinct reduced forms (on the real field, in orthonormal basis) for elements in O(n). - Christian Devanz, Feb 13 2019 a(n) is the number of non-isomorphic asymmetric graphs that can be created by adding a single edge to a path on n+4 vertices. - Emma Farnsworth, Natalie Gomez, Herlandt Lino, and Darren Narayan, Jul 03 2019 a(n+1) is the number of integer triangles with maximum side-length n. - James East, Oct 30 2019 a(n) is the number of nonempty subsets of {1,2,...,n} that contain exactly one odd and one even number. For example, for n=7, a(7)=12 and the 12 subsets are {1,2}, {1,4}, {1,6}, {2,3}, {2,5}, {2,7}, {3,4}, {3,6}, {4,5}, {4,7}, {5,6}, {6,7}. - Enrique Navarrete, Dec 16 2019 a(n+1) is also the n-th term of the Saind sequence (w_n)_{n>=1}, i.e., the infinite sequence caused by the entries of the queue of the degree sequences associated with the Saind arrays, as n increases. - Giulia Palma, Jun 24 2020 Aside from the first two terms, a(n) enumerates the number of distinct normal ordered terms in the expansion of the differential operator (x + d/dx)^m associated to the Hermite polynomials and the Heisenberg-Weyl algebra. It also enumerates the number of distinct monomials in the bivariate polynomials corresponding to the partial sums of the series for cos(x+y) and sin(x+y). Cf. A344678. - Tom Copeland, May 27 2021 a(n) is the maximal number of negative products a_i * a_j (1 <= i <= j <= n), where all a_i are real numbers. - Logan Pipes, Jul 08 2021 a(n) is the maximum product of the chromatic numbers of a graph of order n-1 and its complement. The extremal graphs are characterized in the papers of Finck (1968) and Bickle (2023).
a(n) is the maximum product of the degeneracies of a graph of order n+1 and its complement. The extremal graphs are characterized in the paper of Bickle (2012). (End)
a(n) is the maximum number m such that m white rooks and m black rooks can coexist on an n-1 X n-1 chessboard without attacking each other. - Aaron Khan, Jul 13 2022 a(n) is the number of 231-avoiding odd Grassmannian permutations of size n. - Juan B. Gil, Mar 10 2023 a(n) is the number of integer tuples (x,y) satisfying n + x + y >= 0, 25*n + x - 11*y >=0, 25*n - 11*x + y >=0, n + x + y == 0 (mod 12) , 25*n + x - 11*y == 0 (mod 5), 25*n - 11*x + y == 0 (mod 5) . For n=2, the sole solution is (x,y) = (0,0) and so a(2) = 1. For n = 3, the a(3) = 2 solutions are (-3, 2) and (2, -3). - Jeffery Opoku, Feb 16 2024 Let us consider triangles whose vertices are the centers of three squares constructed on the sides of a right triangle. a(n) is the integer part of the area of these triangles, taken without repetitions and in ascending order. See the illustration in the links. - Nicolay Avilov, Aug 05 2024 For n>=2, a(n) is the indendence number of the 2-token graph F_2(P_n) of the path graph P_n on n vertices. (Alternatively, as noted by Peter Munn, F_2(P_n) is the nXn square lattice, or grid, graph diminished by a cut across the diagonal.) - Miquel A. Fiol, Oct 05 2024 For n >= 1, also the lower matching number of the n-triangular honeycomb rook graph. - Eric W. Weisstein, Dec 14 2024
REFERENCES
Sergei Abramovich, Combinatorics of the Triangle Inequality: From Straws to Experimental Mathematics for Teachers, Spreadsheets in Education (eJSiE), Vol. 9, Issue 1, Article 1, 2016. See Fig. 3.
G. L. Alexanderson et al., The William Powell Putnam Mathematical Competition - Problems and Solutions: 1965-1984, M.A.A., 1985; see Problem A-1 of 27th Competition.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.
Michael Doob, The Canadian Mathematical Olympiad -- L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society -- Société Mathématique du Canada, Problème 9, 1970, pp 22-23, 1993.
H. J. Finck, On the chromatic numbers of a graph and its complement. Theory of Graphs (Proc. Colloq., Tihany, 1966) Academic Press, New York (1968), 99-113.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 99.
D. E. Knuth, The art of programming, Vol. 1, 3rd Edition, Addison-Wesley, 1997, Ex. 36 of section 1.2.4.
J. Nelder, Critical sets in Latin squares, CSIRO Division of Math. and Stats. Newsletter, Vol. 38 (1977), p. 4.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
P. J. Cameron, BCC Problem List, Problem BCC15.15 (DM285), Discrete Math., Vol. 167/168 (1997), pp. 605-615. J. W. L. Glaisher and G. Carey Foster, The Method of Quarter Squares, Journal of the Institute of Actuaries, Vol. 28, No. 3, January, 1890, pp. 227-235. W. Mantel and W. A. Wythoff, Vraagstuk XXVIII, Wiskundige Opgaven, Vol. 10 (1907), pp. 60-61.
FORMULA
a(n) = (2*n^2-1+(-1)^n)/8. - Paul Barry, May 27 2003 G.f.: x^2/((1-x)^2*(1-x^2)) = x^2 / ( (1+x)*(1-x)^3 ). - Simon Plouffe in his 1992 dissertation, leading zeros dropped E.g.f.: exp(x)*(2*x^2+2*x-1)/8 + exp(-x)/8.
a(-n) = a(n) for all n in Z.
a(n) = a(n-1) + a(n-2) - a(n-3) + 1 [with a(-1) = a(0) = a(1) = 0], a(2k) = k^2, a(2k-1) = k(k-1). - Henry Bottomley, Mar 08 2000 0*0, 0*1, 1*1, 1*2, 2*2, 2*3, 3*3, 3*4, ... with an obvious pattern.
a(n) = Sum_{k=1..n} floor(k/2). - Yong Kong (ykong(AT)curagen.com), Mar 10 2001
a(n) = n*floor((n-1)/2) - floor((n-1)/2)*(floor((n-1)/2)+ 1); a(n) = a(n-2) + n-2 with a(1) = 0, a(2) = 0. - Santi Spadaro, Jul 13 2001 Also: a(n) = binomial(n, 2) - a(n-1) = A000217(n-1) - a(n-1) with a(0) = 0. - Labos Elemer, Apr 26 2003 a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k, 2). - Paul Barry, Jul 01 2003 a(n) = (-1)^n * partial sum of alternating triangular numbers. - Jon Perry, Dec 30 2003 a(n) = a(n-2) + n - 1, n > 1. - Paul Barry, Jul 14 2004 a(n+1) = Sum_{i=0..n} min(i, n-i). - Marc LeBrun, Feb 15 2005 a(n+1) = Sum_{k = 0..floor((n-1)/2)} n-2k; a(n+1) = Sum_{k=0..n} k*(1-(-1)^(n+k-1))/2. - Paul Barry, Apr 16 2005 1 + 1/(1 + 2/(1 + 4/(1 + 6/(1 + 9/(1 + 12/(1 + 16/(1 + ...))))))) = 6/(Pi^2 - 6) = 1.550546096730... - Philippe Deléham, Jun 20 2005 Sequence starting (2, 2, 4, 6, 9, ...) = A128174 (as an infinite lower triangular matrix) * vector [1, 2, 3, ...]; where A128174 = (1; 0,1; 1,0,1; 0,1,0,1; ...). - Gary W. Adamson, Jul 27 2007 a(n) = Sum_{i=k..n} P(i, k) where P(i, k) is the number of partitions of i into k parts. - Thomas Wieder, Sep 01 2007 a(n) = round((2*n^2-1)/8) = round(n^2/4) = ceiling((n^2-1)/4). - Mircea Merca, Nov 29 2010 n*a(n+2) = 2*a(n+1) + (n+2)*a(n). Holonomic Ansatz with smallest order of recurrence. - Thotsaporn Thanatipanonda, Dec 12 2010 a(n) = floor(b(n)) with b(n) = b(n-1) + n/(1+e^(1/n)) and b(0)= 0. - Richard R. Forberg, Jun 08 2013 a(n) = floor((n+2)/2 - 1)*(floor((n+2)/2)-1 + (n+2) mod 2). - Wesley Ivan Hurt, Jun 09 2013 0 = a(n)*a(n+2) + a(n+1)*(-2*a(n+2) + a(n+3)) for all integers n. - Michael Somos, Nov 22 2014 a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n-1)/2). - Wesley Ivan Hurt, Mar 12 2015 a(n) = a(n-3) + floor(3*n/2) - 2. - Yuchun Ji, Aug 14 2020 Sum_{n>=2} (-1)^n/a(n) = Pi^2/6 - 1. - Amiram Eldar, Mar 10 2022
EXAMPLE
a(3) = 2, floor(3/2)*ceiling(3/2) = 2.
[ n] a(n)
---------
[ 2] 1
[ 3] 2
[ 4] 1 + 3
[ 5] 2 + 4
[ 6] 1 + 3 + 5
[ 7] 2 + 4 + 6
[ 8] 1 + 3 + 5 + 7
[ 9] 2 + 4 + 6 + 8
Tiling of a triangular shape T_N, N >= 1 with rectangles:
N=5, n=6: a(6) = 9 because all the rectangles (i, j) (modulo transposition, i.e., interchange of i and j) which are of use are:
(5, 1) ; (1, 1)
(4, 2), (4, 1) ; (2, 2), (2, 1)
; (3, 3), (3, 2), (3, 1)
That is (1+1) + (2+2) + 3 = 9 = a(6). Partial sums of 1, 1, 2, 2, 3, ... ( A004526). (End) Bisymmetric matrices B: 2 X 2, a(3) = 2 from B[1,1] and B[1,2]. 3 X 3, a(4) = 4 from B[1,1], B[1,2], B[1,3], and B[2,2]. - Wolfdieter Lang, Jul 07 2015 Letting n=5, there are a(n)=a(5)=6 partitions of 2n+1=11 of length three with exactly two even entries:
(8,2,1) |- 2n+1
(7,2,2) |- 2n+1
(6,4,1) |- 2n+1
(6,3,2) |- 2n+1
(5,4,2) |- 2n+1
(4,4,3) |- 2n+1
(End)
Examples of the sequence when used for rooks on a chessboard:
.
A solution illustrating a(5)=4:
+---------+
| B B . . |
| B B . . |
| . . W W |
| . . W W |
+---------+
.
A solution illustrating a(6)=6:
+-----------+
| B B . . . |
| B B . . . |
| B B . . . |
| . . W W W |
| . . W W W |
+-----------+
(End)
MAPLE
A002620 := n->floor(n^2/4); G002620 := series(x^2/((1-x)^2*(1-x^2)), x, 60);
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card<r), U=Sequence(Z, card>=1)}, unlabeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m), m=0..57) ; # Zerinvary Lajos, Mar 09 2007
MATHEMATICA
LinearRecurrence[{2, 0, -2, 1}, {0, 0, 1, 2}, 60] (* Harvey P. Dale, Oct 05 2012 *) CoefficientList[Series[-(x^2/((-1 + x)^3 (1 + x))), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 11 2018 *)
PROG
(Magma) [ Floor(n/2)*Ceiling(n/2) : n in [0..40]];
(PARI) a(n)=n^2\4
(PARI) (t(n)=n*(n+1)/2); for(i=1, 50, print1(", ", (-1)^i*sum(k=1, i, (-1)^k*t(k))))
(PARI) x='x+O('x^100); concat([0, 0], Vec(x^2/((1-x)^2*(1-x^2)))) \\ Altug Alkan, Oct 15 2015 (Haskell)
(Maxima) makelist(floor(n^2/4), n, 0, 50); /* Martin Ettl, Oct 17 2012 */ (Sage)
x, y = 0, 1
yield x
while true:
yield x
x, y = x + y, x//y + 1
(GAP) # using the formula by Paul Barry
(Python)
CROSSREFS
A087811 is another version of this sequence.
Cf. A024206, A072280, A002984, A007590, A000212, A118015, A056827, A118013, A128174, A000601, A115514, A189151, A063657, A171608, A005044, A030179, A275437, A004526. Antidiagonal sums of array A003983. Elliptic troublemaker sequences: A000212 (= R_n(1,3) = R_n(2,3)), A007590 (= R_n(2,4)), A030511 (= R_n(2,6) = R_n(4,6)), A033436 (= R_n(1,4) = R_n(3,4)), A033437 (= R_n(1,5) = R_n(4,5)), A033438 (= R_n(1,6) = R_n(5,6)), A033439 (= R_n(1,7) = R_n(6,7)), A184535 (= R_n(2,5) = R_n(3,5)). Cf. A250000 (queens on a chessboard), A176222 (kings on a chessboard), A355509 (knights on a chessboard).
Numbers not divisible by 3.
(Formerly M0957 N0357)
+10
204
1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104
COMMENTS
Earliest monotonic sequence starting with (1,2) and satisfying the condition: "a(n)+a(n-1) is not in the sequence." - Benoit Cloitre, Mar 25 2004. [The numbers of the form a(n)+a(n-1) form precisely the complement with respect to the positive integers. - David W. Wilson, Feb 18 2012] a(1) = 1; a(n) is least number which is relatively prime to the sum of all the previous terms. - Amarnath Murthy, Jun 18 2001 Notice the property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 3). - Bruno Berselli, Nov 17 2010 The set of natural numbers (1, 2, 3, ...), sequence A000027; represents the numbers of ordered compositions of n using terms in the signed set: (1, 2, -4, -5, 7, 8, -10, -11, 13, 14, ...). This follows from (1, 2, 3, ...) being the INVERT transform of A011655, signed and beginning: (1, 1, 0, -1, -1, 0, 1, 1, 0, ...). - Gary W. Adamson, Apr 28 2013 Numbers whose sum of digits (and digital root) is != 0 (mod 3). - Joerg Arndt, Aug 29 2014 The number of partitions of 3*(n-1) into at most 2 parts. - Colin Barker, Apr 22 2015 a(n) is the number of partitions of 3*n into two distinct parts. - L. Edson Jeffery, Jan 14 2017 Conjectured (and like even easily proved) to be the graph bandwidth of the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017 Numbers k such that Fibonacci(k) mod 4 = 1 or 3. Equivalently, sequence lists the indices of the odd Fibonacci numbers (see A014437). - Bruno Berselli, Oct 17 2017 Minimum value of n_3 such that the "rectangular spiral pattern" is the optimal solution for Ripà's n_1 X n_2 x n_3 Dots Problem, for any n_1 = n_2. For example, if n_1 = n_2 = 5, n_3 = floor((3/2)*(n_1 - 1)) + 1 = a(5). - Marco Ripà, Jul 23 2018 For n >= 54, a(n) = sat(n, P_n), the minimum number of edges in a P_n-saturated graph on n vertices, where P_n is the n-vertex path (see Dudek, Katona, and Wojda, 2003; Frick and Singleton, 2005). - Danny Rorabaugh, Nov 07 2017 a(n) is the smallest sum of arch lengths for the top arches of a semi-meander with n arches. An arch length is the number of arches covered + 1.
/\ The top arch has a length of 3. /\ The top arch has a length of 3.
/ \ Both bottom arches have a //\\ The middle arch has a length of 2.
//\/\\ length of 1. ///\\\ The bottom arch has a length of 1.
Example: a(6) = 8 /\ /\
//\\ /\ //\\ /\ 2 + 1 + 1 + 2 + 1 + 1 = 8. (End)
This is the lexicographically earliest increasing sequence of positive integers such that no polynomial of degree d can be fitted to d+2 consecutive terms (equivalently, such that no iterated difference is zero). - Pontus von Brömssen, Dec 26 2021
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Aneta Dudek, Gyula Y. Katona, and A.Pawel Wojda, m_Path Cover Saturated Graphs, Electronic Notes in Discrete Math., Vol. 13 (April 2003), pp. 41-44. Brian Hopkins, Euler's Enumerations, Enumerative Combinatorics and Applications, Vol. 1, No. 1 (2021), Article #S1H1.
FORMULA
a(n) = 3 + a(n-2) for n > 2.
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
a(2*n+1) = 3*n+1, a(2*n) = 3*n-1.
G.f.: x * (1 + x + x^2) / ((1 - x) * (1 - x^2)). - Michael Somos, Jun 08 2000 a(n) = (4-n)*a(n-1) + 2*a(n-2) + (n-3)*a(n-3) (from the Carlitz et al. article).
a(1) = 1, a(n) = 2*a(n-1) - 3*floor(a(n-1)/3). - Benoit Cloitre, Aug 17 2002 Nearest integer to (Sum_{k>=n} 1/k^3)/(Sum_{k>=n} 1/k^4). - Benoit Cloitre, Jun 12 2003 Euler transform of length 3 sequence [2, 1, -1]. - Michael Somos, Sep 06 2008 a(n) = 3*floor(n/2) + (-1)^(n+1). - Gary Detlefs, Dec 29 2011 1/1^3 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + ... = 4 Pi^3/(3 sqrt(3)). - M. F. Hasler, Mar 29 2015 E.g.f.: (4 + sinh(x) - cosh(x) + 3*(2*x - 1)*exp(x))/4. - Ilya Gutkovskiy, May 24 2016 a(n) = a(n+k-1) + a(n-k) - a(n-1) for n > k >= 0. - Bob Selcoe, Feb 03 2017 Product_{n>=1} (1 - (-1)^n/a(n)) = 1.
Product_{n>=2} (1 + (-1)^n/a(n)) = 2*Pi/(3*sqrt(3)) ( A248897). (End)
EXAMPLE
G.f.: x + 2*x^2 + 4*x^3 + 5*x^4 + 7*x^5 + 8*x^6 + 10*x^7 + 11*x^8 + 13*x^9 + ...
MAPLE
a[1]:=1:a[2]:=2:for n from 3 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=1..69); # Zerinvary Lajos, Mar 16 2008, offset corrected by M. F. Hasler, Apr 07 2015
MATHEMATICA
Drop[Range[200 + 1], {1, -1, 3}] - 1 (* József Konczer, May 24 2016 *) CoefficientList[Series[(x^2 + x + 1)/((x - 1)^2 (x + 1)), {x, 0, 70}],
x] (* or *)
PROG
(PARI) x='x+O('x^100); Vec(x*(1+x+x^2)/((1-x)*(1-x^2))) \\ Altug Alkan, Oct 22 2015 (Haskell)
a001651 = (`div` 2) . (subtract 1) . (* 3)
a001651_list = filter ((/= 0) . (`mod` 3)) [1..]
(GAP) Filtered([0..110], n->n mod 3<>0); # Muniru A Asiru, Jul 24 2018 (Python)
(Python)
CROSSREFS
Cf. A000726, A001082, A003105, A005408 (n=1 or 3 mod 4), A007494, A008585 (complement), A011655, A026386, A032766, A073010, A191967, A225227, A004526, A248897. Cf. A000027, A000217, A000292, A000982, A001477, A008619, A014437, A040001, A047239, A047257, A077043, A084858, A113801, A141425, A215879.
EXTENSIONS
This is a list, so the offset should be 1. I corrected this and adjusted some of the comments and formulas. Other lines probably also need to be adjusted. - N. J. A. Sloane, Jan 01 2011
Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3) + n + 1.
(Formerly M1100 N0419)
+10
79
1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988, 1160, 1351, 1562, 1794, 2048, 2325, 2626, 2952, 3304, 3683, 4090, 4526, 4992, 5489, 6018, 6580, 7176, 7807, 8474, 9178, 9920, 10701, 11522, 12384, 13288, 14235, 15226
COMMENTS
Note that a(n) = a(n-1) + A000124(n-1). This has the following geometrical interpretation: Define a number of planes in space to be in general arrangement when (1) no two planes are parallel,
(2) there are no two parallel intersection lines,
(3) there is no point common to four or more planes.
Suppose there are already n-1 planes in general arrangement, thus defining the maximal number of regions in space obtainable by n-1 planes and now one more plane is added in general arrangement. Then it will cut each of the n-1 planes and acquire intersection lines which are in general arrangement. (See the comments on A000124 for general arrangement with lines.) These lines on the new plane define the maximal number of regions in 2-space definable by n-1 straight lines, hence this is A000124(n-1). Each of this regions acts as a dividing wall, thereby creating as many new regions in addition to the a(n-1) regions already there, hence a(n) = a(n-1) + A000124(n-1). - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006 More generally, we have: A000027(n) = binomial(n,0) + binomial(n,1) (the natural numbers), A000124(n) = binomial(n,0) + binomial(n,1) + binomial(n,2) (the Lazy Caterer's sequence), a(n) = binomial(n,0) + binomial(n,1) + binomial(n,2) + binomial(n,3) (Cake Numbers). - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006 If Y is a 2-subset of an n-set X then, for n>=3, a(n-3) is the number of 3-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007 a(n) is the number of compositions (ordered partitions) of n+1 into four or fewer parts or equivalently the sum of the first four terms in the n-th row of Pascal's triangle. - Geoffrey Critzer, Jan 23 2009 a(n) is also the maximum number of different values obtained by summing n consecutive positive integers with all possible 2^n sign combinations. This maximum is first reached when summing the interval [n, 2n-1]. - Olivier Gérard, Mar 22 2010 a(n) contains only 5 perfect squares > 1: 4, 64, 576, 67600, and 75203584. The incidences of > 0 are given by A047694. - Frank M Jackson, Mar 15 2013 Given n tiles with two values - an A value and a B value - a player may pick either the A value or the B value. The particular tiles are [n, 0], [n-1, 1], ..., [2, n-2] and [1, n-1]. The sequence is the number of different final A:B counts. For example, with n=4, we can have final total [5, 3] = [4, _] + [_, 1] + [_, 2] + [1, _] = [_, 0] + [3, _] + [2, _] + [_, 3], so a(4) = 2^4 - 1 = 15. The largest and smallest final A+B counts are given by A077043 and A002620 respectively. - Jon Perry, Oct 24 2014 For n>=3, a(n) is also the number of maximal cliques in the (n+1)-triangular graph (the 4-triangular graph has a(3)=8 maximal cliques). - Andrew Howroyd, Jul 19 2017 a(n) is the number of binary words of length n matching the regular expression 1*0*1*0*. Coincidentally, A000124 counts binary words of the form 0*1*0*. See Alexandersson and Nabawanda for proof. - Per W. Alexandersson, May 15 2021 For n > 0, let the n-dimensional cube, {0,1}^n be provided with the Hamming distance, d. Given an element x in {0,1}^n, a(n) is the number of elements y in {0,1}^n such that d(x, y) <= 3. Example: n = 4. Let x = (0,0,0,0) be in {0,1}^4.
d(x,y) = 0: y in {(0,0,0,0)}.
d(x,y) = 1: y in {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)}.
d(x,y) = 2: y in {(1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0), (0,1,0,1), (0,0,1,1)}.
d(x,y) = 3: y in {(1,1,1,0), (1,1,0,1), (1,0,1,1), (0,1,1,1)}.
All these y are at a distance <= 3 from (0,0,0,0), so a(4) = 15. (See Peter C. Heinig's formula). - Yosu Yurramendi, Dec 14 2021 For n >= 2, a(n) is the number of distinct least squares regression lines fitted to n points (j,y_j), 1 <= j <= n, where each y_j is 0 or 1. The number of distinct lines with exactly k 1's among y_1, ..., y_n is A077028(n,k). The number of distinct slopes is A123596(n). - Pontus von Brömssen, Mar 16 2024 The only powers of 2 in this sequence are a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8, and a(7) = 64. - Jianing Song, Jan 02 2025
REFERENCES
V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_3.
R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 27.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 177.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. H. Stickels, Mindstretching Puzzles. Sterling, NY, 1994 p. 85.
W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 30.
A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #45 (First published: San Francisco: Holden-Day, Inc., 1964)
LINKS
Sebastian Mizera and Sabrina Pasterski, Celestial Geometry, arXiv:2204.02505 [hep-th], 2022.
FORMULA
a(n) = (n+1)*(n^2-n+6)/6 = (n^3 + 5*n + 6) / 6.
G.f.: (1 - 2*x + 2x^2)/(1-x)^4. - [ Simon Plouffe in his 1992 dissertation.] E.g.f.: (1 + x + x^2/2 + x^3/6)*exp(x).
a(n) = binomial(n,3) + binomial(n,2) + binomial(n,1) + binomial(n,0). - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
Paraphrasing the previous comment: the sequence is the binomial transform of [1,1,1,1,0,0,0,...]. - Gary W. Adamson, Oct 23 2007 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
Inverse binomial transform of A134396. Sum_{n>=0} a(n)/n! = 8*exp(1)/3. (End)
EXAMPLE
a(4)=15 because there are 15 compositions of 5 into four or fewer parts. a(6)=42 because the sum of the first four terms in the 6th row of Pascal's triangle is 1+6+15+20=42. - Geoffrey Critzer, Jan 23 2009 For n=5, (1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 35) and their opposite are the 26 different sums obtained by summing 5,6,7,8,9 with any sign combination. - Olivier Gérard, Mar 22 2010 G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 26*x^5 + 42*x^6 + 64*x^7 + ... - Michael Somos, Jul 07 2022
MATHEMATICA
Table[(n^3 + 5 n + 6)/6, {n, 0, 50}] (* Harvey P. Dale, Jan 19 2013 *) LinearRecurrence[{4, -6, 4, -1}, {1, 2, 4, 8}, 50] (* Harvey P. Dale, Jan 19 2013 *)
PROG
(PARI) Vec((1-2*x+2*x^2)/((1-x)^4) + O(x^100)) \\ Altug Alkan, Oct 16 2015 (Python)
def A000125_gen(): # generator of terms a, b, c = 1, 1, 1
while True:
yield a
a, b, c = a+b, b+c, c+1
CROSSREFS
Cf. A000124, A003600, A005408, A016813, A086514, A058331, A002522, A161701- A161705, A000127, A161706- A161708, A080856, A161710- A161713, A161715, A006261, A063865, A051601, A077043, A002620, A123596.
Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.
+10
77
1, 4, 11, 20, 33, 48, 67, 88, 113, 140, 171, 204, 241, 280, 323, 368, 417, 468, 523, 580, 641, 704, 771, 840, 913, 988, 1067, 1148, 1233, 1320, 1411, 1504, 1601, 1700, 1803, 1908, 2017, 2128, 2243, 2360, 2481, 2604, 2731, 2860, 2993, 3128, 3267
COMMENTS
In the following guide to related sequences: M=max(x,y,z), m=min(x,y,z), and R=range=M-m. In some cases, it is an offset of the listed sequence which fits the conditions shown for w,x,y. Each sequence satisfies a linear recurrence relation, some of which are identified in the list by the following code (signature):
A: 2, 0, -2, 1, i.e., a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4);
B: 3, -2, -2, 3, -1;
C: 4, -6, 4, -1;
D: 1, 2, -2, -1, 1;
E: 2, 1, -4, 1, 2, -1;
F: 2, -1, 1, -2, 1;
G: 2, -1, 0, 1, -2, 1;
H: 2, -1, 2, -4, 2, -1, 2, -1;
I: 3, -3, 2, -3, 3, -1;
J: 4, -7, 8, -7, 4, -1.
...
A212959 ... |w-x|=|x-y| ...... recurrence type A
A212960 ... |w-x| != |x-y| ................... B
A212683 ... |w-x| < |x-y| .................... B
A212684 ... |w-x| >= |x-y| ................... B
A212963 ... see entry for definition ......... B
A212964 ... |w-x| < |x-y| < |y-w| ............ B
A006331 ... |w-x| < y ........................ C
A005900 ... |w-x| <= y ....................... C
A212965 ... w = R ............................ D
A212967 ... w < R ............................ E
A212968 ... w >= R ........................... E
A077043 ... w = x > R ........................ A
A212969 ... w != x and x > R ................. E
A212970 ... w != x and x < R ................. E
A011934 ... w < floor((x+y)/2) ............... B
A182260 ... w > floor((x+y)/2) ............... B
A055232 ... w <= floor((x+y)/2) .............. B
A011934 ... w >= floor((x+y)/2) .............. B
A212971 ... w < floor((x+y)/3) ............... B
A212972 ... w >= floor((x+y)/3) .............. B
A212973 ... w <= floor((x+y)/3) .............. B
A212974 ... w > floor((x+y)/3) ............... B
A212975 ... R is even ........................ E
A212976 ... R is odd ......................... E
A212980 ... w < x + y and x < y .............. B
A212981 ... w <= x+y and x < y ............... B
A212982 ... w < x + y and x <= y ............. B
A212983 ... w <= x + y and x <= y ............ B
A002623 ... w >= x + y and x <= y ............ B
A087811 ... w = 2*x + y ...................... A
A008805 ... w = 2*x + 2*y .................... D
A000982 ... 2*w = x + y ...................... F
A001318 ... 2*w = 2*x + y .................... F
A008810 ... 3*x = 2*x + y .................... F
A001972 ... w = 4*x + y ...................... G
A212988 ... 4*w = x + y ...................... G
A016061 ... n < w + x + y <= 2*n ............. C
A000292 ... w + x + y <=n .................... C
A000292 ... 2*n < w + x + y <= 3*n ........... C
A143785 ... w < R < x ........................ E
A005996 ... w < R <= x ....................... E
A128624 ... w <= R <= x ...................... E
A213041 ... R = 2*|w - x| .................... A
A213045 ... R < 2*|w - x| .................... B
A087035 ... R >= 2*|w - x| ................... B
A213388 ... R <= 2*|w - x| ................... B
A171218 ... M < 2*m .......................... B
A213389 ... R < 2|w - x| ..................... E
A213390 ... M >= 2*m ......................... E
A213391 ... 2*M < 3*m ........................ H
A213392 ... 2*M >= 3*m ....................... H
A213393 ... 2*M > 3*m ........................ H
A213391 ... 2*M <= 3*m ....................... H
A047838 ... w = |x + y - w| .................. A
A213396 ... 2*w < |x + y - w| ................ I
A213397 ... 2*w >= |x + y - w| ............... I
A000384 ... max(|w-x|,|x-y|) = |w-y|
A213398 ... min(|w-x|,|x-y|) = x ............. A
A213399 ... max(|w-x|,|x-y|) = x ............. D
A213479 ... max(|w-x|,|x-y|) = w+x+y ......... D
A213480 ... max(|w-x|,|x-y|) != w+x+y ........ E
A006918 ... |w-x| + |x-y| > w+x+y ............ E
A213481 ... |w-x| + |x-y| <= w+x+y ........... E
A213482 ... |w-x| + |x-y| < w+x+y ............ E
A213483 ... |w-x| + |x-y| >= w+x+y ........... E
A213484 ... |w-x|+|x-y|+|y-w| = w+x+y
A213485 ... |w-x|+|x-y|+|y-w| != w+x+y ....... J
A213486 ... |w-x|+|x-y|+|y-w| > w+x+y ........ J
A213487 ... |w-x|+|x-y|+|y-w| >= w+x+y ....... J
A213488 ... |w-x|+|x-y|+|y-w| < w+x+y ........ J
A213489 ... |w-x|+|x-y|+|y-w| <= w+x+y ....... J
A213490 ... w,x,y,|w-x|,|x-y| distinct
A213491 ... w,x,y,|w-x|,|x-y| not distinct
A213493 ... w,x,y,|w-x|,|x-y|,|w-y| distinct
A213495 ... w = min(|w-x|,|x-y|,|w-y|)
A213492 ... w != min(|w-x|,|x-y|,|w-y|)
A213498 ... w != max(|w-x|,|x-y|,|w-y|)
...
A211795 includes a guide for sequences that count 4-tuples (w,x,y,z) having all terms in {0,...,n} and satisfying selected properties. Some of the sequences indexed at A211795 satisfy recurrences that are represented in the above list.
REFERENCES
A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.
FORMULA
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1+2*x+3*x^2)/((1+x)*(1-x)^3).
a(n) = (6*n^2 + 8*n + 3 + (-1)^n)/4. - Luce ETIENNE, Apr 05 2014
EXAMPLE
a(1)=4 counts these (x,y,z): (0,0,0), (1,1,1), (0,1,0), (1,0,1).
Numbers congruent to {1, 3} mod 6: 1, 3, 7, 9, 13, 15, 19, ...
a(0) = 1;
a(1) = 1 + 3 = 4;
a(2) = 1 + 3 + 7 = 11;
a(3) = 1 + 3 + 7 + 9 = 20;
a(4) = 1 + 3 + 7 + 9 + 13 = 33;
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] == Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 50]] (* A212959 *)
Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e., A000217(n) + A002620(n)).
(Formerly M3329)
+10
37
0, 1, 4, 8, 14, 21, 30, 40, 52, 65, 80, 96, 114, 133, 154, 176, 200, 225, 252, 280, 310, 341, 374, 408, 444, 481, 520, 560, 602, 645, 690, 736, 784, 833, 884, 936, 990, 1045, 1102, 1160, 1220, 1281, 1344, 1408, 1474, 1541, 1610, 1680, 1752, 1825, 1900, 1976, 2054
COMMENTS
Equals (1, 2, 3, 4, ...) convolved with (1, 2, 1, 2, ...). a(4) = 14 = (1, 2, 3, 4) dot (2, 1, 2, 1) = (2 + 2 + 6 + 4). - Gary W. Adamson, May 01 2009 We observe that is the transform of A032766 by the following transform T: T(u_0,u_1,u_2,u_3,...) = (u_0, u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In other words, v_p = Sum_{k=0..p} u_k and the g.f. phi_v of is given by phi_v = phi_u/(1-z). - Richard Choulet, Jan 28 2010 Equals row sums of a triangle with (1, 4, 7, 10, ...) in every column, shifted down twice for columns > 1. - Gary W. Adamson, Mar 03 2010 Number of pairs (x,y) with x in {0,...,n}, y odd in {0,...,2n}, and x < y. - Clark Kimberling, Jul 02 2012
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
Expansion of x*(1+2*x) / ((1-x)^2*(1-x^2)). - Simon Plouffe in his 1992 dissertation a(n) = (6*n^2 + 4*n - 1 + (-1)^n)/8. - Paul Barry, May 30 2003 a(n) = Sum_{i=1..n} (n - i + 1) * 2^( (i+1) mod 2 ). - Wesley Ivan Hurt, Mar 30 2014 Sum_{n>=1} 1/a(n) = 3 - Pi/(4*sqrt(3)) - 3*log(3)/4. - Amiram Eldar, May 28 2022 E.g.f.: (x*(5 + 3*x)*cosh(x) - (1 - 5*x - 3*x^2)*sinh(x))/4. - Stefano Spezia, Aug 22 2023
EXAMPLE
G.f. = x + 4*x^2 + 8*x^3 + 14*x^4 + 21*x^5 + 30*x^6 + 40*x^7 + 52*x^8 + 65*x^9 + ...
MAPLE
with (combinat): seq(count(Partition((3*n+1)), size=3), n=0..52); # Zerinvary Lajos, Mar 28 2008 # 2nd program
(6*n^2 + 4*n - 1 + (-1)^n)/8 ;
MATHEMATICA
Accumulate[LinearRecurrence[{1, 1, -1}, {0, 1, 3}, 100]] (* Harvey P. Dale, Sep 29 2013 *) a[ n_] := Quotient[n + 1, 2] (Quotient[n, 2] 3 + 1); (* Michael Somos, Jun 09 2014 *) a[ n_] := Quotient[3 (n + 1)^2 + 1, 4] - (n + 1); (* Michael Somos, Jun 10 2015 *) LinearRecurrence[{2, 0, -2, 1}, {0, 1, 4, 8}, 53] (* Ray Chandler, Aug 03 2015 *)
PROG
(PARI) {a(n) = (3*(n+1)^2 + 1)\4 - n - 1}; /* Michael Somos, Mar 10 2006 */
Number of balls in pyramid with base either a regular hexagon or a hexagon with alternate sides differing by 1 (balls in hexagonal pyramid of height n taken from hexagonal close-packing).
+10
36
0, 1, 4, 11, 23, 42, 69, 106, 154, 215, 290, 381, 489, 616, 763, 932, 1124, 1341, 1584, 1855, 2155, 2486, 2849, 3246, 3678, 4147, 4654, 5201, 5789, 6420, 7095, 7816, 8584, 9401, 10268, 11187, 12159, 13186
COMMENTS
Alternately add and subtract successively longer sets of integers: 0; 1 = 0+1; -4 = 1-2-3; 11 = -4+4+5+6; -23 = 11-7-8-9-10; 42 = -23+11+12+13+14+15; -69 = 42-16-17-18-19-20-21; ... then take absolute values. - Walter Carlini, Aug 28 2003 Number of 3 X 3 symmetric matrices with nonnegative integer entries, such that every row (and column) sum equals n-1.
Equals Sum_{0..n} of "three-quarter squares" sequence ( A077043). - Philipp M. Buluschek (kitschen(AT)romandie.com), Aug 12 2007 Sum of all the smallest parts in the partitions of 3n into three parts (see example). - Wesley Ivan Hurt, Jan 23 2014 For n > 0, a(n) is the number of (nonnegative integer) magic labelings of the prism graph Y_3 with magic sum n - 1. - L. Edson Jeffery, Sep 09 2017 Or number of magic labelings of LOOP X C_3 with magic sum n - 1, where LOOP is the 1-vertex, 1-loop-edge graph, as Y_k = I X C_k and LOOP X C_k have the same numbers of magic labelings when k is odd. - David J. Seal, Sep 13 2017 a(n) is the number of triples of integers in [1,n]^3 such that each pair has sum larger than n. - Bob Zwetsloot, Jul 23 2020
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Prop. 4.6.21, p. 235, G_3(lambda).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.14(a), p. 452.
FORMULA
a(n) = floor((n^2+1)(2n+3)/8).
G.f.: x*(x^2+x+1)/((x+1)*(x-1)^4).
a(n) = floor((2n^3 + 3n^2 + 2n)/8); also nearest integer to ((n+1)^4 - n^4)/16.
a(n) = (4n^3 + 6n^2 + 4n+1 - (-1)^n)/16. - Wesley Petty (Wesley.Petty(AT)mail.tamucc.edu), Mar 06 2004
a(n) = Sum_{i=1..n} i^2 - floor(i^2/4) = Sum_{i=1..n} i * (2n - 2i + 1 - floor((n - i + 1)/2) ). - Wesley Ivan Hurt, Jan 23 2014 E.g.f.: (1/16)*(-exp(-x) + exp(x)*(1 + 14*x + 18*x^2 + 4*x^3)). - Stefano Spezia, Nov 29 2019 a(2*n) = (1/2)*( n*(n + 1)^3 - (n - 1)*n^3 ); a(2*n-1) = (1/2)*( (n + 1)*n^3 - n*(n - 1)^3 ) (note: replacing the exponent 3 with 2 throughout gives the sequence of generalized pentagonal numbers A001318). - Peter Bala, Aug 11 2021
EXAMPLE
Add last column for a(n) (n > 0).
13 + 1 + 1
12 + 2 + 1
11 + 3 + 1
10 + 4 + 1
9 + 5 + 1
8 + 6 + 1
7 + 7 + 1
10 + 1 + 1 11 + 2 + 2
9 + 2 + 1 10 + 3 + 2
8 + 3 + 1 9 + 4 + 2
7 + 4 + 1 8 + 5 + 2
6 + 5 + 1 7 + 6 + 2
7 + 1 + 1 8 + 2 + 2 9 + 3 + 3
6 + 2 + 1 7 + 3 + 2 8 + 4 + 3
5 + 3 + 1 6 + 4 + 2 7 + 5 + 3
4 + 4 + 1 5 + 5 + 2 6 + 6 + 3
4 + 1 + 1 5 + 2 + 2 6 + 3 + 3 7 + 4 + 4
3 + 2 + 1 4 + 3 + 2 5 + 4 + 3 6 + 5 + 4
1 + 1 + 1 2 + 2 + 2 3 + 3 + 3 4 + 4 + 4 5 + 5 + 5
3(1) 3(2) 3(3) 3(4) 3(5) .. 3n
---------------------------------------------------------------------
1 4 11 23 42 .. a(n)
MAPLE
series(x*(x^2+x+1)/(x+1)/(x-1)^4, x, 80);
MATHEMATICA
Table[ Ceiling[3*n^2/4], {n, 0, 37}] // Accumulate (* Jean-François Alcover, Dec 20 2012, after Philipp M. Buluschek's comment *) CoefficientList[Series[x (x^2 + x + 1) / ((x + 1) (x - 1)^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 28 2013 *) LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 4, 11, 23}, 38] (* L. Edson Jeffery, Sep 09 2017 *)
CROSSREFS
Cf. A061927, A244497, A292281, A244873, A289992 (# of magic labelings of prism graph Y_k = I X C_k, for k = 4,5,6,7,8, up to an offset).
AUTHOR
Eric E Blom (eblom(AT)REM.re.uokhsc.edu)
EXTENSIONS
Error in n=8 term corrected May 15 1997
Triangular numbers plus quarter-squares: n*(n+1)/2 + floor((n+1)^2/4) (i.e., A000217(n) + A002620(n+1)).
(Formerly M1368 N0531)
+10
25
0, 2, 5, 10, 16, 24, 33, 44, 56, 70, 85, 102, 120, 140, 161, 184, 208, 234, 261, 290, 320, 352, 385, 420, 456, 494, 533, 574, 616, 660, 705, 752, 800, 850, 901, 954, 1008, 1064, 1121, 1180, 1240, 1302, 1365, 1430, 1496, 1564, 1633, 1704, 1776, 1850, 1925
COMMENTS
Number of series-reduced planted trees with n+7 nodes and 3 internal nodes.
The trees enumerated with 3 internal nodes are of two types. Those with all internal nodes at different heights are enumerated by the triangular numbers. Those with two internal nodes at the same height are enumerated by the quarter squares. - Michael Somos, May 19 2000 Number of pairs (x,y) with x in {0,...,n}, y even in {0,...,2n}, and x < y. - Clark Kimberling, Jul 02 2012
REFERENCES
John Riordan, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = n + floor( (3n^2+1)/4 ).
G.f.: (2*x+x^2)/((1-x)^2*(1-x^2)).
a(n) = (6n^2 + 8n + 1 - (-1)^n)/8;
a(n) = Sum_{k=0..n} max(k, n-k). (End)
Starting (2, 5, 10, 16, 24, ...), = binomial transform of [2, 3, 2, -1, 2, -4, 8, -16, 32, ...]. - Gary W. Adamson, Nov 30 2007 a(0)=0, a(1)=2, a(2)=5, a(3)=10, a(n) = 2*a(n-1) + 0*a(n-2) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Apr 01 2012 0 = -6 + a(n) - 2*a(n+2) + a(n+4) for all n in Z. - Michael Somos, Nov 03 2014 0 = a(n)*(+a(n+1) - a(n+2)) + a(n+1)*(-3 - a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Nov 03 2014 Sum_{n>=1} 1/a(n) = 3/4 - Pi/(4*sqrt(3)) + 3*log(3)/4. - Amiram Eldar, May 28 2022 E.g.f.: (x*(7 + 3*x)*cosh(x) + (1 + 7*x + 3*x^2)*sinh(x))/4. - Stefano Spezia, Aug 22 2023
EXAMPLE
For n=1 we find 2 planted trees with 8 nodes, 3 of which are internal (i) and 5 are endpoints (e):
.e...e...e...e....e...e....
...i.......i........i...e..
.......i..............i...e
.......e................i..
........................e..
G.f. = 2*x + 5*x^2 + 10*x^3 + 16*x^4 + 24*x^5 + 33*x^6 + 44*x^7 + 56*x^8 + ...
MAPLE
A001859:=(-1-z^2-2*z^3+z^4)/(z+1)/(z-1)^3; # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence with an additional leading 1
with (combinat):seq(count(Partition((3*n+2)), size=3), n=0..50); # Zerinvary Lajos, Mar 28 2008
MATHEMATICA
With[{nn=60}, Total/@Thread[{Accumulate[Range[0, nn]], Floor[Range[ nn+1]^2/4]}]] (* or *) LinearRecurrence[{2, 0, -2, 1}, {0, 2, 5, 10}, 60] (* Harvey P. Dale, Apr 01 2012 *)
PROG
(PARI) {a(n) = n + (3*n^2 + 1) \ 4};
(Haskell)
CROSSREFS
Antidiagonal sums of array A003984.
Concentric pentagonal numbers: floor( 5*n^2 / 4 ).
+10
24
0, 1, 5, 11, 20, 31, 45, 61, 80, 101, 125, 151, 180, 211, 245, 281, 320, 361, 405, 451, 500, 551, 605, 661, 720, 781, 845, 911, 980, 1051, 1125, 1201, 1280, 1361, 1445, 1531, 1620, 1711, 1805, 1901, 2000, 2101, 2205, 2311, 2420, 2531, 2645, 2761, 2880, 3001
COMMENTS
a(n) = -N(-floor(n/2),n) with the N(a,b) = ((2*a+b)^2 - b^2*5)/4, the norm for integers a + b*omega(5), a, b rational integers, in the quadratic number field Q(sqrt(5)), where omega(5) = (1 + sqrt(5))/2 (golden section).
a(n) = max({|N(a,n)|,a = -n..+n}) = |N(-floor(n/2),n)| = n^2 + n*floor(n/2) - floor(n/2)^2 = floor(5*n^2/4) (the last eq. checks for even and odd n). (End)
FORMULA
a(n) = 5*n^2/4+((-1)^n-1)/8. - Omar E. Pol, Sep 28 2011 G.f.: x*(1+3*x+x^2)/(1-2*x+2*x^3-x^4). - Colin Barker, Jan 06 2012 a(n) = a(-n); a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>0, a(-1) = 1, a(0) = 0, a(1) = 1, a(2) = 5, n >= 3. (See the Bruno Berselli recurrence and a general comment for primes 1 (mod 4) under A227541). - Wolfdieter Lang, Aug 08 2013 a(n) = Sum_{j=1..n} Sum{i=1..n} ceiling((i+j-n+1)/2). - Wesley Ivan Hurt, Mar 12 2015 Sum_{n>=1} 1/a(n) = Pi^2/30 + tan(Pi/(2*sqrt(5)))*Pi/sqrt(5). - Amiram Eldar, Jan 16 2023
EXAMPLE
Illustration of initial terms (In a precise representation the pentagons should appear strictly concentric):
.
. o
. o o
. o o o o
. o o o o o o
. o o o o o o o o o
. o o o o o o o o o o
. o o o o o o o o o o o o o
. o o o o o o o o
. o o o o o o o o o o o o o o o
.
. 1 5 11 20 31
.
(End)
PROG
(Haskell)
a032527 n = a032527_list !! n
a032527_list = scanl (+) 0 a047209_list
(Python)
Square array read by antidiagonals with T(n,k) = k*n^2/4+(k-4)*((-1)^n-1)/8, n>=0, k>=0.
+10
24
0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 3, 2, 1, 0, 1, 4, 5, 3, 1, 0, 0, 7, 8, 7, 4, 1, 0, 1, 9, 13, 12, 9, 5, 1, 0, 0, 13, 18, 19, 16, 11, 6, 1, 0, 1, 16, 25, 27, 25, 20, 13, 7, 1, 0, 0, 21, 32, 37, 36, 31, 24, 15, 8, 1, 0, 1, 25, 41, 48, 49, 45, 37, 28, 17, 9, 1, 0
COMMENTS
Also, if k >= 2 and m = 2*k, then column k lists the numbers of the form k*n^2 and the centered m-gonal numbers interleaved.
For k >= 3, this is also a table of concentric polygonal numbers. Column k lists the concentric k-gonal numbers.
It appears that the first differences of column k are the numbers that are congruent to {1, k-1} mod k, if k >= 3.
EXAMPLE
Array begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...
0, 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
1, 7, 13, 19, 25, 31, 37, 43, 49, 55, ...
0, 9, 18, 27, 36, 45, 54, 63, 72, 81, ...
1, 13, 25, 37, 49, 61, 73, 85, 97, 109, ...
0, 16, 32, 48, 64, 80, 96, 112, 128, 144, ...
1, 21, 41, 61, 81, 101, 121, 141, 161, 181, ...
0, 25, 50, 75, 100, 125, 150, 175, 200, 225, ...
...
MAPLE
k*n^2/4+((-1)^n-1)*(k-4)/8 ;
end proc:
for d from 0 to 12 do
for k from 0 to d do
end do:
MATHEMATICA
t[n_, k_] := k*n^2/4+(k-4)*((-1)^n-1)/8; Flatten[ Table[ t[n-k, k], {n, 0, 11}, {k, 0, n}]] (* Jean-François Alcover, Dec 14 2011 *)
PROG
(GAP) nmax:=13;; T:=List([0..nmax], n->List([0..nmax], k->k*n^2/4+(k-4)*((-1)^n-1)/8));; b:=List([2..nmax], n->OrderedPartitions(n, 2));;
a:=Flat(List([1..Length(b)], i->List([1..Length(b[i])], j->T[b[i][j][2]][b[i][j][1]]))); # Muniru A Asiru, Jul 19 2018
CROSSREFS
Rows n: A000004 (n=0), A000012 (n=1), A001477 (n=2), A005408 (n=3), A008586 (n=4), A016921 (n=5), A008591 (n=6), A017533 (n=7), A008598 (n=8), A215145 (n=9), A008607 (n=10). Columns k: A000035 (k=0), A004652 (k=1), A000982 (k=2), A077043 (k=3), A000290 (k=4), A032527 (k=5), A032528 (k=6), A195041 (k=7), A077221 (k=8), A195042 (k=9), A195142 (k=10), A195043 (k=11), A195143 (k=12), A195045 (k=13), A195145 (k=14), A195046 (k=15), A195146 (k=16), A195047 (k=17), A195147 (k=18), A195048 (k=19), A195148 (k=20), A195049 (k=21), A195149 (k=22), A195058 (k=23), A195158 (k=24).
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