A001971 Nearest integer to n^2/8. (Formerly M0625 N0227)

0, 0, 1, 1, 2, 3, 5, 6, 8, 10, 13, 15, 18, 21, 25, 28, 32, 36, 41, 45, 50, 55, 61, 66, 72, 78, 85, 91, 98, 105, 113, 120, 128, 136, 145, 153, 162, 171, 181, 190, 200, 210, 221, 231, 242, 253, 265, 276, 288, 300, 313, 325, 338, 351, 365, 378, 392, 406, 421, 435, 450

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)

For n >= 4, a(n) = A026810(n+2) - A026810(n-4).

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).

For n >= 6, \n,4\ = A026810(n) - A026810(n-6).

(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

a(2*n+1) = A000217(n), a(2*n) = floor((A000217(n-1)+A000217(n)+1)/2). - Carl R. White, Aug 12 2010

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

The 4th diagonal of A061857?

Kind of an inverse of A261491 (regarding Go).

Cf. A026810 (partitions with greatest part 4), A001400 (partitions in at most 4 parts), A000217 (a(2n+1): triangular numbers n(n+1)/2), A000982 (a(2n): round(n^2/2)).

Sequence in context: A347645 A229172 A056837 * A122493 A284830 A053873

Adjacent sequences: A001968 A001969 A001970 * A001972 A001973 A001974

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Edited Feb 08 2004

Status

approved