A074148 -id:A074148

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A211422

Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.

+10
106

1, 9, 17, 25, 41, 49, 57, 65, 81, 105, 113, 121, 137, 145, 153, 161, 193, 201, 225, 233, 249, 257, 265, 273, 289, 329, 337, 361, 377, 385, 393, 401, 433, 441, 449, 457, 505, 513, 521, 529, 545, 553, 561, 569, 585, 609, 617, 625, 657, 713, 753, 761

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Suppose that S={-n,...,0,...,n} and that f(w,x,y,n) is a function, where w,x,y are in S. The number of ordered triples (w,x,y) satisfying f(w,x,y,n)=0, regarded as a function of n, is a sequence t of nonnegative integers. Sequences such as t/4 may also be integer sequences for all except certain initial values of n. In the following guide, such sequences are indicated in the related sequences column and may be included in the corresponding Mathematica programs.

...

sequence... f(w,x,y,n) ..... related sequences

A211415 ... w^2+x*y-1 ...... t+2, t/4, (t/4-1)/4

A211422 ... w^2+x*y ........ (t-1)/8, A120486

A211423 ... w^2+2x*y ....... (t-1)/4

A211424 ... w^2+3x*y ....... (t-1)/4

A211425 ... w^2+4x*y ....... (t-1)/4

A211426 ... 2w^2+x*y ....... (t-1)/4

A211427 ... 3w^2+x*y ....... (t-1)/4

A211428 ... 2w^2+3x*y ...... (t-1)/4

A211429 ... w^3+x*y ........ (t-1)/4

A211430 ... w^2+x+y ........ (t-1)/2

A211431 ... w^3+(x+y)^2 .... (t-1)/2

A211432 ... w^2-x^2-y^2 .... (t-1)/8

A003215 ... w+x+y .......... (t-1)/2, A045943

A202253 ... w+2x+3y ........ (t-1)/2, A143978

A211433 ... w+2x+4y ........ (t-1)/2

A211434 ... w+2x+5y ........ (t-1)/4

A211435 ... w+4x+5y ........ (t-1)/2

A211436 ... 2w+3x+4y ....... (t-1)/2

A211435 ... 2w+3x+5y ....... (t-1)/2

A211438 ... w+2x+2y ....... (t-1)/2, A118277

A001844 ... w+x+2y ......... (t-1)/4, A000217

A211439 ... w+3x+3y ........ (t-1)/2

A211440 ... 2w+3x+3y ....... (t-1)/2

A028896 ... w+x+y-1 ........ t/6, A000217

A211441 ... w+x+y-2 ........ t/3, A028387

A182074 ... w^2+x*y-n ...... t/4, A028387

A000384 ... w+x+y-n

A000217 ... w+x+y-2n

A211437 ... w*x*y-n ........ t/4, A007425

A211480 ... w+2x+3y-1

A211481 ... w+2x+3y-n

A211482 ... w*x+w*y+x*y-w*x*y

A211483 ... (n+w)^2-x-y

A182112 ... (n+w)^2-x-y-w

...

For the following sequences, S={1,...,n}, rather than

{-n,...,0,...n}. If f(w,x,y,n) is linear in w,x,y,n, then the sequence is a linear recurrence sequence.

A132188 ... w^2-x*y

A211506 ... w^2-x*y-n

A211507 ... w^2-x*y+n

A211508 ... w^2+x*y-n

A211509 ... w^2+x*y-2n

A211510 ... w^2-x*y+2n

A211511 ... w^2-2x*y ....... t/2

A211512 ... w^2-3x*y ....... t/2

A211513 ... 2w^2-x*y ....... t/2

A211514 ... 3w^2-x*y ....... t/2

A211515 ... w^3-x*y

A211516 ... w^2-x-y

A211517 ... w^3-(x+y)^2

A063468 ... w^2-x^2-y^2 .... t/2

A000217 ... w+x-y

A001399 ... w-2x-3y

A211519 ... w-2x+3y

A008810 ... w+2x-3y

A001399 ... w-2x-3y

A008642 ... w-2x-4y

A211520 ... w-2x+4y

A211521 ... w+2x-4y

A000115 ... w-2x-5y

A211522 ... w-2x+5y

A211523 ... w+2x-5y

A211524 ... w-3x-5y

A211533 ... w-3x+5y

A211523 ... w+3x-5y

A211535 ... w-4x-5y

A211536 ... w-4x+5y

A008812 ... w+4x-5y

A055998 ... w+x+y-2n

A074148 ... 2w+x+y-2n

A211538 ... 2w+2x+y-2n

A211539 ... 2w+2x-y-2n

A211540 ... 2w-3x-4y

A211541 ... 2w-3x+4y

A211542 ... 2w+3x-4y

A211543 ... 2w-3x-5y

A211544 ... 2w-3x+5y

A008812 ... 2w+3x-5y

A008805 ... w-2x-2y (repeated triangular numbers)

A001318 ... w-2x+2y

A000982 ... w+x-2y

A211534 ... w-3x-3y

A211546 ... w-3x+3y (triply repeated triangular numbers)

A211547 ... 2w-3x-3y (triply repeated squares)

A082667 ... 2w-3x+3y

A055998 ... w-x-y+2

A001399 ... w-2x-3y+1

A108579 ... w-2x-3y+n

...

Next, S={-n,...-1,1,...,n}, and the sequence counts the cases (w,x,y) satisfying the indicated inequality. If f(w,x,y,n) is linear in w,x,y,n, then the sequence is a linear recurrence sequence.

A211545 ... w+x+y>0; recurrence degree: 4

A211612 ... w+x+y>=0

A211613 ... w+x+y>1

A211614 ... w+x+y>2

A211615 ... |w+x+y|<=1

A211616 ... |w+x+y|<=2

A211617 ... 2w+x+y>0; recurrence degree: 5

A211618 ... 2w+x+y>1

A211619 ... 2w+x+y>2

A211620 ... |2w+x+y|<=1

A211621 ... w+2x+3y>0

A211622 ... w+2x+3y>1

A211623 ... |w+2x+3y|<=1

A211624 ... w+2x+2y>0; recurrence degree: 6

A211625 ... w+3x+3y>0; recurrence degree: 8

A211626 ... w+4x+4y>0; recurrence degree: 10

A211627 ... w+5x+5y>0; recurrence degree: 12

A211628 ... 3w+x+y>0; recurrence degree: 6

A211629 ... 4w+x+y>0; recurrence degree: 7

A211630 ... 5w+x+y>0; recurrence degree: 8

A211631 ... w^2>x^2+y^2; all terms divisible by 8

A211632 ... 2w^2>x^2+y^2; all terms divisible by 8

A211633 ... w^2>2x^2+2y^2; all terms divisible by 8

...

Next, S={1,...,n}, and the sequence counts the cases (w,x,y) satisfying the indicated relation.

A211634 ... w^2<=x^2+y^2

A211635 ... w^2<x^2+y^2; see Comments at A211790

A211636 ... w^2>=x^2+y^2

A211637 ... w^2>x^2+y^2

A211638 ... w^2+x^2+y^2<n

A211639 ... w^2+x^2+y^2<=n

A211640 ... w^2+x^2+y^2>n

A211641 ... w^2+x^2+y^2>=n

A211642 ... w^2+x^2+y^2<2n

A211643 ... w^2+x^2+y^2<=2n

A211644 ... w^2+x^2+y^2>2n

A211645 ... w^2+x^2+y^2>=2n

A211646 ... w^2+x^2+y^2<3n

A211647 ... w^2+x^2+y^2<=3n

A063691 ... w^2+x^2+y^2=n

A211649 ... w^2+x^2+y^2=2n

A211648 ... w^2+x^2+y^2=3n

A211650 ... w^3<x^3+y^3; see Comments at A211790

A211651 ... w^3>x^3+y^3; see Comments at A211790

A211652 ... w^4<x^4+y^4; see Comments at A211790

A211653 ... w^4>x^4+y^4; see Comments at A211790

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..10000

EXAMPLE

a(1) counts these 9 triples: (-1,-1,1), (-1, 1,-1), (0, -1, 0), (0, 0, -1), (0,0,0), (0,0,1), (0,1,0), (1,-1,1), (1,1,-1).

MATHEMATICA

t[n_] := t[n] = Flatten[Table[w^2 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]

c[n_] := Count[t[n], 0]

t = Table[c[n], {n, 0, 70}] (* A211422 *)

(t - 1)/8 (* A120486 *)

CROSSREFS

Cf. A120486.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Apr 10 2012

STATUS

approved

A109613

Odd numbers repeated.

+10
83

1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25, 27, 27, 29, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 39, 41, 41, 43, 43, 45, 45, 47, 47, 49, 49, 51, 51, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 63, 63, 65, 65, 67, 67, 69, 69, 71, 71, 73

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The number of rounds in a round-robin tournament with n competitors. - A. Timothy Royappa, Aug 13 2011

Diagonal sums of number triangle A113126. - Paul Barry, Oct 14 2005

When partitioning a convex n-gon by all the diagonals, the maximum number of sides in resulting polygons is 2*floor(n/2)+1 = a(n-1) (from Moscow Olympiad problem 1950). - Tanya Khovanova, Apr 06 2008

The inverse values of the coefficients in the series expansion of f(x) = (1/2)*(1+x)*log((1+x)/(1-x)) lead to this sequence; cf. A098557. - Johannes W. Meijer, Nov 12 2009

From Reinhard Zumkeller, Dec 05 2009: (Start)

First differences: A010673; partial sums: A000982;

A059329(n) = Sum_{k = 0..n} a(k)*a(n-k);

A167875(n) = Sum_{k = 0..n} a(k)*A005408(n-k);

A171218(n) = Sum_{k = 0..n} a(k)*A005843(n-k);

A008794(n+2) = Sum_{k = 0..n} a(k)*A059841(n-k). (End)

Dimension of the space of weight 2n+4 cusp forms for Gamma_0(5). - Michael Somos, May 29 2013

For n > 4: a(n) = A230584(n) - A230584(n-2). - Reinhard Zumkeller, Feb 10 2015

The arithmetic function v+-(n,2) as defined in A290988. - Robert Price, Aug 22 2017

For n > 0, also the chromatic number of the (n+1)-triangular (Johnson) graph. - Eric W. Weisstein, Nov 17 2017

a(n-1), for n >= 1, is also the upper bound a_{up}(b), where b = 2*n + 1, in the first (top) row of the complete coach system Sigma(b) of Hilton and Pedersen [H-P]. All odd numbers <= a_{up}(b) of the smallest positive restricted residue system of b appear once in the first rows of the c(2*n+1) = A135303(n) coaches. If b is an odd prime a_{up}(b) is the maximum. See a comment in the proof of the quasi-order theorem of H-P, on page 263 ["Furthermore, every possible a_i < b/2 ..."]. For an example see below. - Wolfdieter Lang, Feb 19 2020

Satisfies the nested recurrence a(n) = a(a(n-2)) + 2*a(n-a(n-1)) with a(0) = a(1) = 1. Cf. A004001. - Peter Bala, Aug 30 2022

The binomial transform is 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560,.. (see A057711). - R. J. Mathar, Feb 25 2023

REFERENCES

Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, 3rd printing 2012, pp. (260-281).

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Chromatic Number

Eric Weisstein's World of Mathematics, Johnson Graph

Eric Weisstein's World of Mathematics, Triangular Graph

Wikipedia, Round-robin tournament

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

FORMULA

a(n) = 2*floor(n/2) + 1.

a(n) = A052928(n) + 1 = 2*A004526(n) + 1.

a(n) = A028242(n) + A110654(n).

a(n) = A052938(n-2) + A084964(n-2) for n > 1. - Reinhard Zumkeller, Aug 27 2005

G.f.: (1 + x + x^2 + x^3)/(1 - x^2)^2. - Paul Barry, Oct 14 2005

a(n) = 2*a(n-2) - a(n-4), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 3. - Philippe Deléham, Nov 03 2008

a(n) = A001477(n) + A059841(n). - Philippe Deléham, Mar 31 2009

a(n) = 2*n - a(n-1), with a(0) = 1. - Vincenzo Librandi, Nov 13 2010

a(n) = R(n, -2), where R(n, x) is the n-th row polynomial of A211955. a(n) = (-1)^n + 2*Sum_{k = 1..n} (-1)^(n - k - 2)*4^(k-1)*binomial(n+k, 2*k). Cf. A084159. - Peter Bala, May 01 2012

a(n) = A182579(n+1, n). - Reinhard Zumkeller, May 06 2012

G.f.: ( 1 + x^2 ) / ( (1 + x)*(x - 1)^2 ). - R. J. Mathar, Jul 12 2016

E.g.f.: x*exp(x) + cosh(x). - Ilya Gutkovskiy, Jul 12 2016

From Guenther Schrack, Sep 10 2018: (Start)

a(-n) = -a(n-1).

a(n) = A047270(n+1) - (2*n + 2).

a(n) = A005408(A004526(n)). (End)

a(n) = A000217(n) / A004526(n+1), n > 0. - Torlach Rush, Nov 10 2023

EXAMPLE

G.f. = 1 + x + 3*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 7*x^6 + 7*x^7 + 9*x^8 + 9*x^9 + ...

Complete coach system for (a composite) b = 2*n + 1 = 33: Sigma(33) ={[1; 5], [5, 7, 13; 2, 1, 2]} (the first two rows are here 1 and 5, 7, 13), a_{up}(33) = a(15) = 15. But 15 is not in the reduced residue system modulo 33, so the maximal (odd) a number is 13. For the prime b = 31, a_{up}(31) = a(14) = 15 appears as maximum of the first rows. - Wolfdieter Lang, Feb 19 2020

MAPLE

A109613:=n->2*floor(n/2)+1; seq(A109613(k), k=0..100); # Wesley Ivan Hurt, Oct 22 2013

MATHEMATICA

Flatten@ Array[{2# - 1, 2# - 1} &, 37] (* Robert G. Wilson v, Jul 07 2012 *)

(# - Boole[EvenQ[#]] &) /@ Range[80] (* Alonso del Arte, Sep 11 2019 *)

With[{c=2*Range[0, 40]+1}, Riffle[c, c]] (* Harvey P. Dale, Jan 02 2020 *)

PROG

(Haskell)

a109613 = (+ 1) . (* 2) . (`div` 2)

a109613_list = 1 : 1 : map (+ 2) a109613_list

-- Reinhard Zumkeller, Oct 27 2012, Feb 21 2011

(PARI) A109613(n)=n>>1<<1+1 \\ Charles R Greathouse IV, Feb 24 2011

(Sage) def a(n) : return( len( CuspForms( Gamma0( 5), 2*n + 4, prec=1). basis())); # Michael Somos, May 29 2013

(Scala) ((1 to 49) by 2) flatMap { List.fill(2)(_) } // Alonso del Arte, Sep 11 2019

CROSSREFS

Cf. A000217, A063196, A110660, A186421, A186422, A211955, A230584, A290988.

Complement of A052928 with respect to the universe A004526. - Guenther Schrack, Aug 21 2018

First differences of A000982, A061925, A074148, A105343, A116940, and A179207. - Guenther Schrack, Aug 21 2018

KEYWORD

nonn,easy

AUTHOR

Reinhard Zumkeller, Aug 01 2005

STATUS

approved

A047838

a(n) = floor(n^2/2) - 1.

+10
22

1, 3, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479, 511, 543, 577, 611, 647, 683, 721, 759, 799, 839, 881, 923, 967, 1011, 1057, 1103, 1151, 1199, 1249, 1299, 1351, 1403

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,2

COMMENTS

Define the organization number of a permutation pi_1, pi_2, ..., pi_n to be the following. Start at 1, count the steps to reach 2, then the steps to reach 3, etc. Add them up. Then the maximal value of the organization number of any permutation of [1..n] for n = 0, 1, 2, 3, ... is given by 0, 1, 3, 7, 11, 17, 23, ... (this sequence). This was established by Graham Cormode (graham(AT)research.att.com), Aug 17 2006, see link below, answering a question raised by Tom Young (mcgreg265(AT)msn.com) and Barry Cipra, Aug 15 2006

From Dmitry Kamenetsky, Nov 29 2006: (Start)

This is the length of the longest non-self-intersecting spiral drawn on an n X n grid. E.g., for n=5 the spiral has length 17:

1 0 1 1 1

1 0 1 0 1

1 0 0 0 1

1 1 1 1 1 (End)

It appears that a(n+1) is the maximum number of consecutive integers (beginning with 1) that can be placed, one after another, on an n-peg Towers of Hanoi, such that the sum of any two consecutive integers on any peg is a square. See the problem: http://online-judge.uva.es/p/v102/10276.html. - Ashutosh Mehra, Dec 06 2008

a(n) = number of (w,x,y) with all terms in {0,...,n} and w = |x+y-w|. - Clark Kimberling, Jun 11 2012

The same sequence also represents the solution to the "pigeons problem": maximal value of the sum of the lengths of n-1 line segments (connected at their end-points) required to pass through n trail dots, with unit distance between adjacent points, visiting all of them without overlaping two or more segments. In this case, a(0)=0, a(1)=1, a(2)=3, and so on. - Marco Ripà, Jan 28 2014

Also the longest path length in the n X n white bishop graph. - Eric W. Weisstein, Mar 27 2018

a(n) is the number of right triangles with sides n*(h-floor(h)), floor(h) and h, where h is the hypotenuse. - Andrzej Kukla, Apr 14 2021

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Laurent Bulteau, Samuele Giraudo and Stéphane Vialette, Disorders and permutations , 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Article No. 18; pp. 18:1-18:14.

Graham Cormode, Notes on the organization number of a permutation.

Eric Weisstein's World of Mathematics, Longest Path Problem.

Eric Weisstein's World of Mathematics, White Bishop Graph.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

FORMULA

a(2)=1; for n > 2, a(n) = a(n-1) + n - 1 + (n-1 mod 2). - Benoit Cloitre, Jan 12 2003

a(n) = T(n-1) + floor(n/2) - 1 = T(n) - floor((n+3)/2), where T(n) is the n-th triangular number (A000217). - Robert G. Wilson v, Aug 31 2006

Equals (n-1)-th row sums of triangles A134151 and A135152. Also, = binomial transform of [1, 2, 2, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Nov 21 2007

G.f.: x^2*(1+x+x^2-x^3)/((1-x)^3*(1+x)). - R. J. Mathar, Sep 09 2008

a(n) = floor((n^2 + 4*n + 2)/2). - Gary Detlefs, Feb 10 2010

a(n) = abs(A188653(n)). - Reinhard Zumkeller, Apr 13 2011

a(n) = (2*n^2 + (-1)^n - 5)/4. - Bruno Berselli, Sep 14 2011

a(n) = a(-n) = A007590(n) - 1.

a(n) = A080827(n) - 2. - Kevin Ryde, Aug 24 2013

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n > 4. - Wesley Ivan Hurt, Aug 06 2015

a(n) = A000217(n-1) + A004526(n-2), for n > 1. - J. Stauduhar, Oct 20 2017

From Guenther Schrack, May 12 2018: (Start)

Set a(0) = a(1) = -1, a(n) = a(n-2) + 2*n - 2 for n > 1.

a(n) = A000982(n-1) + n - 2 for n > 1.

a(n) = 2*A033683(n) - 3 for n > 1.

a(n) = A061925(n-1) + n - 3 for n > 1.

a(n) = A074148(n) - n - 1 for n > 1.

a(n) = A105343(n-1) + n - 4 for n > 1.

a(n) = A116940(n-1) - n for n > 1.

a(n) = A179207(n) - n + 1 for n > 1.

a(n) = A183575(n-2) + 1 for n > 2.

a(n) = A265284(n-1) - 2*n + 1 for n > 1.

a(n) = 2*A290743(n) - 5 for n > 1. (End)

E.g.f.: 1 + x + ((x^2 + x - 2)*cosh(x) + (x^2 + x - 3)*sinh(x))/2. - Stefano Spezia, May 06 2021

Sum_{n>=2} 1/a(n) = 3/2 + tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)) - cot(Pi/sqrt(2))*Pi/(2*sqrt(2)). - Amiram Eldar, Sep 15 2022

EXAMPLE

x^2 + 3*x^3 + 7*x^4 + 11*x^5 + 17*x^6 + 23*x^7 + 31*x^8 + 39*x^9 + 49*x^10 + ...

MAPLE

seq(floor((n^2+4*n+2)/2), n=0..20) # Gary Detlefs, Feb 10 2010

MATHEMATICA

Table[Floor[n^2/2] - 1, {n, 2, 60}] (* Robert G. Wilson v, Aug 31 2006 *)

LinearRecurrence[{2, 0, -2, 1}, {1, 3, 7, 11}, 60] (* Harvey P. Dale, Jan 16 2015 *)

Floor[Range[2, 20]^2/2] - 1 (* Eric W. Weisstein, Mar 27 2018 *)

Table[((-1)^n + 2 n^2 - 5)/4, {n, 2, 20}] (* Eric W. Weisstein, Mar 27 2018 *)

CoefficientList[Series[(-1 - x - x^2 + x^3)/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 27 2018 *)

PROG

(PARI) a(n) = n^2\2 - 1

(Magma) [Floor(n^2/2)-1 : n in [2..100]]; // Wesley Ivan Hurt, Aug 06 2015

CROSSREFS

Complement of A047839. First difference is A052928.

Cf. A000217, A007590, A080827, A134151, A135151, A135152, A188653.

Partial sums: A213759(n-1) for n > 1. - Guenther Schrack, May 12 2018

KEYWORD

nonn,easy

AUTHOR

Michael Somos, May 07 1999

EXTENSIONS

Edited by Charles R Greathouse IV, Apr 23 2010

STATUS

approved

A061925

a(n) = ceiling(n^2/2) + 1.

+10
18

1, 2, 3, 6, 9, 14, 19, 26, 33, 42, 51, 62, 73, 86, 99, 114, 129, 146, 163, 182, 201, 222, 243, 266, 289, 314, 339, 366, 393, 422, 451, 482, 513, 546, 579, 614, 649, 686, 723, 762, 801, 842, 883, 926, 969, 1014, 1059, 1106, 1153, 1202, 1251, 1302, 1353, 1406

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n+1) gives index of the first occurrence of n in A100795. - Amarnath Murthy, Dec 05 2004

First term in each group in A074148. - Amarnath Murthy, Aug 28 2002

From Christian Barrientos, Jan 01 2021: (Start)

For n >= 3, a(n) is the number of square polyominoes with at least 2n - 2 cells whose bounding box has size 2 X n.

For n = 3, there are 6 square polyominoes with a bounding box of size 2 X 3:

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

|_|_|_| |_|_|_| |_|_|_| |_|_|_| |_|_|_| |_|_|_

|_|_|_| |_|_| |_| |_| |_| |_| |_|_|

(End)

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..1000

Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

FORMULA

a(n) = a(n-1) + 2*floor((n-1)/2) + 1 = A061926(3, k) = 2*A002620(n+1) - (n-1) = A000982(n) + 1.

a(2*n) = a(2*n-1) + 2*n - 1 = 2*n^2 + 1 = A058331(n).

a(2*n+1) = a(2*n) + 2*n + 1 = 2*(n^2 + n + 1) = A051890(n+1).

a(n) = floor((n^2+3)/2). - Gary Detlefs, Feb 13 2010

From R. J. Mathar, Feb 19 2010: (Start)

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).

G.f.: (1-x^2+2*x^3)/((1+x) * (1-x)^3). (End)

a(n) = (2*n^2 - (-1)^n + 5)/4. - Bruno Berselli, Sep 29 2011

a(n) = A007590(n+1) - n + 1. - Wesley Ivan Hurt, Jul 15 2013

a(n) + a(n+1) = A027688(n). a(n+1) - a(n) = A109613(n). - R. J. Mathar, Jul 20 2013

E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x))/2. - Stefano Spezia, May 07 2021

MAPLE

seq(floor((n^2+3)/2), n=0..25); # Gary Detlefs, Feb 13 2010

MATHEMATICA

Table[Ceiling[n^2/2]+1, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)

LinearRecurrence[{2, 0, -2, 1}, {1, 2, 3, 6}, 60] (* Harvey P. Dale, Jan 03 2024 *)

PROG

(PARI) { for (n=0, 1000, write("b061925.txt", n, " ", ceil(n^2/2) + 1) ) } \\ Harry J. Smith, Jul 29 2009

CROSSREFS

Cf. A100795, A074147-A074149.

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, May 17 2001

EXTENSIONS

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 09 2007

STATUS

approved

A099392

a(n) = floor((n^2 - 2*n + 3)/2).

+10
16

1, 1, 3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Guenther Schrack, Table of n, a(n) for n = 1..10010

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

FORMULA

a(n) = ceiling(n^2/2)-n+1. - Paul Barry, Jul 16 2006; index shifted by R. J. Mathar, Jul 29 2007

a(n) = ceiling(A002522(n-1)/2). - Branko Curgus, Sep 02 2007

From R. J. Mathar, Feb 20 2011: (Start)

G.f.: x *( -1+x-x^2-x^3 ) / ( (1+x)*(x-1)^3 ).

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).

a(n+1) = (3 + 2*n^2 + (-1)^n)/4. (End)

a(n) = A007590(n-1) + 1 for n >= 2. - Richard R. Forberg, Aug 01 2013

a(n) = A000217(n) - A007494(n-1). - Bui Quang Tuan, Mar 27 2015

From Guenther Schrack, Apr 17 2018: (Start)

a(n) = (2*n^2 - 4*n + 5 -(-1)^n)/4.

a(n+2) = a(n) + 2*n for n > 0.

a(n) = 2*A033683(n-1) - 1 for n > 0.

a(n) = A047838(n-1) + 2 for n > 2.

a(n) = A074148(n-1) - n + 2 for n > 1.

a(n) = A183575(n-3) + 3 for n > 3.

a(n) = 2*A290743(n-1) - 3 for n > 0.

a(n) = 2*A290743(n-2) + A109613(n-5) for n > 4.

a(n) = A074148(n) - A014601(n-1) for n > 0. (End)

Sum_{n>=1} 1/a(n) = tanh(Pi/2)*Pi/2 + coth(Pi/sqrt(2))*Pi/(2*sqrt(2)) + 1/2. - Amiram Eldar, Sep 16 2022

E.g.f.: ((2 - x + x^2)*cosh(x) + (3 - x + x^2)*sinh(x) - 2)/2. - Stefano Spezia, Jan 28 2024

MATHEMATICA

Array[Floor[(#^2 - 2 # + 3)/2] &, 54] (* or *)

Rest@ CoefficientList[Series[x (-1 + x - x^2 - x^3)/((1 + x) (x - 1)^3), {x, 0, 54}], x] (* Michael De Vlieger, Apr 21 2018 *)

PROG

(PARI) a(n)=(n^2+3)\2-n \\ Charles R Greathouse IV, Aug 01 2013

CROSSREFS

Differs from A085913 at n = 61. Apart from leading term, identical to A080827.

Cf. A000217, A001844, A002522, A007494, A007590, A058331 (bisections).

From Guenther Schrack, Apr 17 2018: (Start)

First differences: A052928.

Partial sums: A212964(n) + n for n > 0.

Also A058331 and A001844 interleaved. (End)

Cf. A014601, A033683, A047838, A074148, A109613, A183575, A290743.

KEYWORD

nonn,easy

AUTHOR

Ralf Stephan following a suggestion from Luke Pebody, Oct 20 2004

STATUS

approved

A074147

(2n-1) odd numbers followed by 2n even numbers.

+10
15

1, 2, 4, 3, 5, 7, 6, 8, 10, 12, 9, 11, 13, 15, 17, 14, 16, 18, 20, 22, 24, 19, 21, 23, 25, 27, 29, 31, 26, 28, 30, 32, 34, 36, 38, 40, 33, 35, 37, 39, 41, 43, 45, 47, 49, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 62, 64, 66, 68, 70, 72

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Ivan Neretin, Table of n, a(n) for n = 1..5050

FORMULA

T(n,k) = A061925(n-1)+2k, 0<=k<n. - R. J. Mathar, Jul 17 2007, corrected Nov 02 2023

a(n) = A116941(n-1)+1. - Robert G. Wilson v, Mar 09 2017

EXAMPLE

As a triangular array:

2, 4,

3, 5, 7,

6, 8, 10, 12,

9, 11, 13, 15, 17,

14, 16, 18, 20, 22, 24,

...

MAPLE

A074147 := proc(n, k)

ceil((n-1)^2/2)+1+2*k ;

end proc:

seq(seq( A074147(n, k), k=0..n-1) , n=1..12) ; # R. J. Mathar, Nov 02 2023

MATHEMATICA

Flatten@Table[Ceiling[(n - 1)^2/2] + 2 k - 1, {n, 12}, {k, n}] (* Ivan Neretin, Dec 15 2016 *)

CROSSREFS

Cf. A074148, A074149 (row sums), A001614.

KEYWORD

nonn,easy,tabl

AUTHOR

Amarnath Murthy, Aug 28 2002

STATUS

approved

A188236

T(n,k)=Number of nondecreasing arrangements of n numbers in -(n+k-2)..(n+k-2) with sum zero and not more than two numbers equal

+10
15

1, 1, 2, 1, 3, 4, 1, 4, 7, 15, 1, 5, 12, 30, 58, 1, 6, 17, 52, 119, 245, 1, 7, 24, 81, 221, 527, 1082, 1, 8, 31, 121, 374, 1019, 2395, 5020, 1, 9, 40, 172, 598, 1818, 4818, 11376, 24040, 1, 10, 49, 234, 903, 3047, 8964, 23522, 55368, 118154, 1, 11, 60, 311, 1317, 4859

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Table starts

......1......1......1.......1.......1.......1.......1........1........1

......2......3......4.......5.......6.......7.......8........9.......10

......4......7.....12......17......24......31......40.......49.......60

.....15.....30.....52......81.....121.....172.....234......311......403

.....58....119....221.....374.....598.....903....1317.....1852.....2540

....245....527...1019....1818....3047....4859....7435....10994....15791

...1082...2395...4818....8964...15696...26123...41748....64370....96346

...5020..11376..23522...45225...81981..141519..234413...374820...581280

..24040..55368.117209..231596..432491..769915.1316060..2171675..3475284

.118154.275735.594789.1202495.2302608.4209720.7395049.12546170.20642874

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..521

EXAMPLE

Some solutions for n=6 k=4

.-7...-8...-5...-8...-5...-7...-7...-3...-8...-8...-6...-6...-5...-7...-6...-5

.-4...-8...-3...-5...-5...-7...-4...-3...-4...-8...-6...-4...-4...-4...-2...-5

..0...-4...-2...-4...-1...-1....1....1....0....0....2...-4....1....0...-1....1

..2....6....0....3....1....0....2....1....3....2....2....2....1....1...-1....1

..3....7....4....7....3....7....2....2....4....7....3....4....3....3....3....4

..6....7....6....7....7....8....6....2....5....7....5....8....4....7....7....4

CROSSREFS

Row 3 is A074148(n+1)

KEYWORD

nonn,tabl

AUTHOR

R. H. Hardin Mar 24 2011

STATUS

approved

A229093

The clubs patterns appearing in n X n coins.

+10
15

0, 0, 1, 2, 4, 6, 9, 12, 17, 22, 27, 34, 41, 48, 57, 66, 75, 86, 97, 108, 121, 134, 147, 162, 177, 192, 209, 226, 243, 262, 281, 300, 321, 342, 363, 386, 409, 432, 457, 482, 507, 534, 561, 588, 617, 646, 675, 706, 737, 768, 801, 834, 867, 902, 937, 972, 1009, 1046

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

On the Japanese TV show "Tsuki no Koibito", a girl told her boyfriend that she saw a heart in 4 coins. Actually there are a total of 6 distinct patterns appearing in 2 X 2 coins in which each pattern consists of a part of the perimeter of each coin and forms a continuous area.

a(n) is the number of clubs patterns appearing in n X n coins. It is also A008810(n-1), except for the third term. The inverse patterns (stars or voids between clubs) is A030511 (except the second term). See illustration in links.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Kival Ngaokrajang, Illustration for initial terms

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

FORMULA

a(n) = ceiling((n-1)^2/3), a(0) = 0, a(4) = 4.

G.f.: x^2*(x^7-2*x^6+x^5-x^4+x^3-x^2-1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Oct 07 2013

MATHEMATICA

CoefficientList[Series[(x^7 - 2 x^6 + x^5 - x^4 + x^3 - x^2 - 1)/((x - 1)^3 (x^2 + x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 08 2013 *)

LinearRecurrence[{2, -1, 1, -2, 1}, {0, 0, 1, 2, 4, 6, 9, 12, 17, 22}, 70] (* Harvey P. Dale, Feb 05 2020 *)

PROG

(PARI) Vec(x^2*(x^7-2*x^6+x^5-x^4+x^3-x^2-1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Oct 08 2013

(PARI) a(n) = ceil((n-1)^2/3) \\ Charles R Greathouse IV, Jan 06 2016

CROSSREFS

Cf. A008810, A030511, A074148 (heart patterns), A227906, A229154.

KEYWORD

nonn,easy

AUTHOR

Kival Ngaokrajang, Sep 13 2013

EXTENSIONS

More terms from Colin Barker, Oct 08 2013

STATUS

approved

A105636

Transform of n^3 by the Riordan array (1/(1-x^2), x).

+10
13

0, 1, 8, 28, 72, 153, 288, 496, 800, 1225, 1800, 2556, 3528, 4753, 6272, 8128, 10368, 13041, 16200, 19900, 24200, 29161, 34848, 41328, 48672, 56953, 66248, 76636, 88200, 101025, 115200, 130816, 147968, 166753, 187272, 209628, 233928, 260281, 288800, 319600

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Recurrence a(n) = a(n-2) + n^3, starting with a(0)=0, a(1)=1. Also, in physics, a(n)/4 is the trace of the spin operator |S_z|^3 for a particle with spin S=n/2. For example, when S=3/2, the S_z eigenvalues are -3/2, -1/2, +1/2, +3/2 and therefore the sum of the absolute values of their 3rd powers is 2*28/8 = a(3)/4. - Stanislav Sykora, Nov 07 2013

Also the number of 3-cycles in the (n+1)-triangular honeycomb queen graph. - Eric W. Weisstein, Jul 14 2017

With zero prepended and offset 1, the sequence starts 0,0,1,8,28,... for n=1,2,3,... Call this b(n). Consider the partitions of n into two parts (p,q). Then b(n) is the total volume of the family of cubes with side length |q - p|. - Wesley Ivan Hurt, Apr 14 2018

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..9999

Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Eric Weisstein's World of Mathematics, Graph Cycle.

Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

FORMULA

G.f.: x*(1+4*x+x^2)/((1+x)*(1-x)^5).

a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6).

a(n) = (2*n^4 + 8*n^3 + 8*n^2 - 1 + (-1)^n)/16.

a(n) = Sum_{k=0..floor((n-1)/2)} (n-2*k)^3.

a(n+1) = Sum_{k=0..n} k^3*(1 - (-1)^(n+k-1))/2.

a(n) = ((((x^2 - (x mod 2) - 4)/4)^2 - (((x^2 - (x mod 2) - 4)/4) mod 2))/8) = floor(((floor(x^2/4) - 1)^2)/8) where x = 2*n + 2. Replace x with 2*n - 1 to obtain A050534(n) = 3*A000332(n+1). Note that a(2*n) = A060300(n)/2 and a(2*n + 1) = A002593(n+1). - Raphie Frank, Jan 30 2014

a(n) = floor(1/(exp(2/n^2) - 1)^2)/2. Also a(n) = A007590(n+1)*A074148(n-1)/2. - Richard R. Forberg, Oct 26 2014

Sum_{n>=1} 1/a(n) = -cot(Pi/sqrt(2))*Pi/sqrt(2) - 1/2. - Amiram Eldar, Aug 25 2022

MATHEMATICA

LinearRecurrence[{4, -5, 0, 5, -4, 1}, {0, 1, 8, 28, 72, 153}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

CoefficientList[Series[x (1 + 4 x + x^2)/((1 + x) (1 - x)^5), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 26 2012 *)

Table[((-1)^n + 2 n^2 (n + 2)^2 - 1)/16, {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)

PROG

(Magma) [(2*n^4+8*n^3+8*n^2-1)/16+(-1)^n/16: n in [0..50]]; // Vincenzo Librandi, Oct 27 2014

(PARI) my(x='x+O('x^99)); concat(0, Vec(x*(1+4*x+x^2)/((1+x)*(1-x)^5))) \\ Altug Alkan, Apr 16 2018

(Sage) [(2*n^4 +8*n^3 +8*n^2 -1+(-1)^n)/16 for n in range(30)] # G. C. Greubel, Dec 16 2018

(GAP) List([0..30], n -> (2*n^4 +8*n^3 +8*n^2 -1+(-1)^n)/16); # G. C. Greubel, Dec 16 2018

CROSSREFS

Cf. A002620, A000292, A000578, A011934, A231303.

Cf. A289705 (4-cycles), A289706 (5-cycles), A289707 (6-cycles).

KEYWORD

nonn,easy

AUTHOR

Paul Barry, Apr 16 2005

STATUS

approved

A209344

T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero with no three beads in a row equal.

+10
13

1, 1, 2, 1, 3, 1, 1, 4, 4, 4, 1, 5, 7, 15, 5, 1, 6, 12, 35, 40, 14, 1, 7, 17, 72, 145, 146, 21, 1, 8, 24, 128, 400, 770, 514, 51, 1, 9, 31, 205, 883, 2698, 4029, 2032, 102, 1, 10, 40, 311, 1724, 7358, 18646, 22739, 8076, 249, 1, 11, 49, 448, 3045, 16968, 62853, 136000

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Table starts

..1....1.....1......1......1.......1.......1........1........1........1

..2....3.....4......5......6.......7.......8........9.......10.......11

..1....4.....7.....12.....17......24......31.......40.......49.......60

..4...15....35.....72....128.....205.....311......448......618......829

..5...40...145....400....883....1724....3045.....5026.....7827....11684

.14..146...770...2698...7358...16968...34720....64942...113288...186906

.21..514..4029..18646..62853..172610..409199...870122..1699831..3104474

.51.2032.22739.136000.563109.1830872.5016681.12099880.26438711.53392286

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..165

FORMULA

Empirical for row n:

n=2: a(k) = 2*a(k-1) - a(k-2).

n=3: a(k) = 2*a(k-1) - 2*a(k-3) + a(k-4).

n=4: a(k) = 3*a(k-1) - 3*a(k-2) + 2*a(k-3) - 3*a(k-4) + 3*a(k-5) - a(k-6).

n=5: a(k) = 2*a(k-1) + a(k-2) - 3*a(k-3) - a(k-4) + a(k-5) + 3*a(k-6) - a(k-7) - 2*a(k-8) + a(k-9).

n=6: a(k) = 4*a(k-1) - 5*a(k-2) + a(k-3) + a(k-4) + a(k-5) + a(k-6) - 5*a(k-7) + 4*a(k-8) - a(k-9).

EXAMPLE

Some solutions for n=6, k=8:

.-4...-4...-4...-8...-7...-6...-6...-8...-7...-8...-7...-7...-8...-8...-8...-4

.-3...-3...-3...-3....0....1....1....0...-2....0....1...-2....3...-8...-4...-4

..5...-1...-4....4...-4...-1....1....1....8....3....0....8...-4...-4....0...-2

.-2....3...-3....1....2....8....6....4...-5....5...-6....1....0....6....7....5

.-1...-1....6....3....3...-5...-6....0....5...-4....8...-7....6....7....3....7

..5....6....8....3....6....3....4....3....1....4....4....7....3....7....2...-2

CROSSREFS

Row 3 is A074148.

KEYWORD

nonn,tabl

AUTHOR

R. H. Hardin, Mar 06 2012

STATUS

approved

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