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

Author

N. J. A. Sloane, Dec 04 2003

Keywords

• nonn

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

• 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.

Crossrefs

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

Programs

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 *)

Formula

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