A089054 Solution to the non-squashing boxes problem (version 1).

1, 2, 4, 8, 14, 23, 36, 54, 78, 109, 149, 199, 262, 339, 434, 548, 686, 849, 1043, 1269, 1535, 1842, 2199, 2607, 3078, 3613, 4225, 4915, 5700, 6581, 7576, 8686, 9934, 11321, 12871, 14585, 16493, 18596, 20925, 23481, 26303, 29392, 32788, 36492, 40553, 44972, 49799

Offset

0,2

Comments

Given n boxes labeled 1..n, such that box i weighs i grams and can support a total weight of i grams; a(n) = number of stacks of boxes that can be formed such that no box is squashed.

Links

Table of n, a(n) for n=0..46.

Amanda Folsom, Youkow Homma, Jun Hwan Ryu, and Benjamin Tong, On a general class of non-squashing partitions, Discrete Mathematics 339 (2016) 1482-1506.

Oystein J. Rodseth, Sloane's box stacking problem, Discrete Math. 306 (2006), no. 16, 2005-2009.

Øystein J. Rødseth and James A. Sellers, Congruences modulo high powers of 2 for Sloane's box stacking function, Australasian Journal of Combinatorics, Volume 44 (2009), Pages 255-263.

N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003; Discrete Math., 294 (2005), 259-274.

Formula

G.f.: (B(x)-x)/(1-x)^2, where B(x) = g.f. for A088567.

Mathematica

max = 50; B[x_] = 1+x/(1-x) + Sum[x^(3 2^(k-1))/Product[(1-x^(2^j)), {j, 0, k}], {k, 1, Log[2, max]}] + O[x]^max;
A[x_] = (B[x]-x)/(1-x)^2;
CoefficientList[A[x], x] (* Jean-François Alcover, Sep 01 2018 *)

Crossrefs

Cf. A000123, A088567, A089055, A090631, A090632. Row sums of A090641.

Sequence in context: A138526 A286522 A201347 * A055291 A091773 A107055

Adjacent sequences: A089051 A089052 A089053 * A089055 A089056 A089057

Keyword

nonn

Author

N. J. A. Sloane, Dec 04 2003

Status

approved