OFFSET
0,5
COMMENTS
Restricted partitions.
a(0) = a(1) = 0; a(n) are the partitions of floor((3*n+3)/2) with 3 distinct numbers of the set {1, ..., n}; partitions of floor((3*n+3)/2)-C and ceiling((3*n+3)/2)+C have equal numbers. - Paul Weisenhorn, Jun 05 2009, corrected by M. F. Hasler, Jun 16 2022
Odd-indexed terms are the triangular numbers, even-indexed terms are the midpoint (rounded up where necessary) of the surrounding odd-indexed terms. - Carl R. White, Aug 12 2010
a(n+2) is the number of points one can surround with n stones in Go (including the points under the stones). - Thomas Dybdahl Ahle, May 11 2014
Corollary of above: a(n) is the number of points one can surround with n+2 stones in Go (excluding the points under the stones). - Juhani Heino, Aug 29 2015
From Washington Bomfim, Jan 13 2021: (Start)
Let \n,m\ be the number of partitions of n into m non-distinct parts.
For n >= 1, \n,4\ = round((n-2)^2/8).
(End)
REFERENCES
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
M. Jeger, Einfuehrung in die Kombinatorik, Klett, 1975, Bd.2, pages 110 ff. [Paul Weisenhorn, Jun 05 2009]
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
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
G. Almkvist, Invariants, mostly old ones, Pacific J. Math. 86 (1980), no. 1, 1-13. MR0586866 (81j:14029)
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
Shalosh B. Ekhad and Doron Zeilberger, In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?, arXiv:1901.08172 [math.CO], 2019.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
D. Vainsencher and A. M. Bruckstein, On isoperimetrically optimal polyforms, Theoretical Computer Science 406.1-2, 2008, pp. 146-159.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).
FORMULA
The listed terms through a(20)=50 satisfy a(n+2) = a(n-2) + n. - John W. Layman, Dec 16 1999
G.f.: x^2 * (1 - x + x^2) / (1 - 2*x + x^2 - x^4 + 2*x^5 - x^6) = x^2 * (1 - x^6) / ((1 - x) * (1 - x^2) * (1 - x^3) * (1 - x^4)). - Michael Somos, Feb 07 2004
a(n) = floor((n^2+4)/8). - Paul Weisenhorn, Jun 05 2009
From Michael Somos, Aug 29 2015: (Start)
Euler transform of length 6 sequence [ 1, 1, 1, 1, 0, -1].
a(n) = a(-n) for all n in Z. (End)
a(2n) = A000982(n). - M. F. Hasler, Jun 16 2022
Sum_{n>=2} 1/a(n) = 2 + Pi^2/12 + tanh(Pi/2)*Pi/2. - Amiram Eldar, Jul 02 2023
MAPLE
A001971:=-(1-z+z**2)/((z+1)*(z**2+1)*(z-1)**3); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation [Note that this "generating function" is Sum_{n >= 0} a(n+2)*z^n, not a(n)*z^n. - M. F. Hasler, Jun 16 2022]
MATHEMATICA
LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 0, 1, 1, 2, 3}, 70] (* Harvey P. Dale, Jan 30 2014 *)
PROG
(PARI) {a(n) = round(n^2 / 8)};
(PARI) apply( {A001971(n)=n^2\/8}, [0..99]) \\ M. F. Hasler, Jun 16 2022
(Magma) [Round(n^2/8): n in [0..60]]; // Vincenzo Librandi, Jun 23 2011
(Haskell)
a001971 = floor . (+ 0.5) . (/ 8) . fromIntegral . (^ 2)
-- Reinhard Zumkeller, May 08 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited Feb 08 2004
STATUS
approved