A090632 Given n boxes labeled 1..n, such that box i weighs 3i 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.

1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 36, 42, 48, 56, 64, 72, 82, 92, 102, 114, 126, 138, 153, 168, 183, 201, 219, 237, 258, 279, 300, 324, 348, 372, 400, 428, 456, 488, 520, 552, 588, 624, 660, 700, 740, 780, 826, 872, 918, 970, 1022, 1074, 1132, 1190, 1248

Offset

0,2

Links

Table of n, a(n) for n=0..56.

Amanda Folsom et al, On a general class of non-squashing partitions, Discrete Mathematics 339.5 (2016): 1482-1506.

Youkow Homma, Jun Hwan Ryu and Benjamin Tong, Sequence non-squashing partitions, Slides from a talk, Jul 24 2014.

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

N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

Formula

More generally, let a_k(n), k > 1, denote the number of stacks of boxes that can be formed such that no box is squashed wherein we have n boxes labeled 1..n such that box i weighs k*i grams and can support a total weight of i grams. Then a_k(n) has g.f. 1/((1-x)^2*Product_{i>=0} (1-x^(k*(k+1)^i))). - George Andrews, James A. Sellers and Vladeta Jovovic, May 26 2005 (corrected May 31 2005)

Maple

p:=1/(1-q)^2/product((1-q^(3*4^i)), i=0..5): s:=series(p, q, 100): for n from 0 to 99 do printf(`%d, `, coeff(s, q, n)) od: # James A. Sellers, Dec 23 2005

Crossrefs

Cf. A089054, A090631.

Bisection of A064986.

Sequence in context: A022794 A025693 A117930 * A022786 A005704 A022782

Adjacent sequences: A090629 A090630 A090631 * A090633 A090634 A090635

Keyword

nonn

Author

N. J. A. Sloane, Dec 13 2003

Extensions

More terms from Vladeta Jovovic, May 22 2005

Further terms from James A. Sellers, Dec 23 2005

Status

approved