A229914 Number of pyramid polycubes of a given volume in dimension 3.

1, 3, 7, 16, 33, 63, 117, 202, 344, 566, 908, 1419, 2206, 3334, 4988, 7378, 10778, 15535, 22281, 31547, 44405, 62011, 85939, 118281, 162136, 220494, 298531, 402163, 539181, 719301, 956287, 1265022, 1667973, 2190934, 2867470, 3739797, 4864163, 6303461, 8146863, 10499087, 13493267, 17293169, 22111954

Offset: 1

Author

Matthieu Deneufchâtel, Oct 03 2013

Keywords

• nonn

Comments

A pyramid polycube is obtained by gluing together horizontal plateaux (parallelepipeds of height 1) in such a way that (0,0,0) belongs to the first plateau and each cell of coordinate (0,b,c) belonging to the first plateau is such that b , c >= 0.

If the cell with coordinates (a,b,c) belongs to the (a+1)-st plateau (a>0), then the cell with coordinates (a-1, b, c) belongs to the a-th plateau.

Links

• C. Carré, N. Debroux, M. Deneufchatel, J.-Ph. Dubernard, C. Hillariet, J.-G. Luque, and O. Mallet, Enumeration of Polycubes and Dirichlet Convolutions, J. Int. Seq. 18 (2015) 15.11.4; also hal-00905889, version 1.

Crossrefs

A001931 is an upper bound.

Formula

The generating function for the numbers of pyramids of height h and volumes v_1 , ... v_h is (n_1-n_2+1) *(n_2-n_3+1) *... *(n_{h-1}-n_h+1) *(x_1^{n_1} * ... x_h^{n_h}) / ((1-x_1^{n_1}) *(1-x_1^{n_1}*x_2^{n_2}) *... *(1-x_1^{n_1}*x_2^{n_2}*...x_h^{n_h})).

This sequence is obtained with x_1 = ... = x_h = p by summing over n_1>=, ... >= n_h>=1 and then over h.