A229915 Number of espalier polycubes of a given volume in dimension 3.

1, 1, 3, 5, 10, 14, 26, 34, 57, 76, 116, 150, 227, 284, 408, 520, 718, 895, 1226, 1508, 2018, 2487, 3248, 3968, 5160, 6235, 7970, 9653, 12179, 14630, 18367, 21924, 27241, 32506, 39985, 47492, 58203, 68752, 83613, 98730, 119269, 140224, 168799, 197758, 236753, 277052, 329867, 384852, 457006, 531500, 628338

Offset: 0

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 with coordinates (0,b,c) belongs to the first plateau 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.

An espalier polycube is a special pyramid such that each plateau contains the cell with coordinates (a,0,0).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..100
• C. Carré, N. Debroux, M. Deneufchatel, J.-P. Dubernard et al., Dirichlet convolution and enumeration of pyramid polycubes, 2013.
• C. Carre, 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.

Crossrefs

Cf. A026820, A100882, A227925, A230118, A229917, A229925, A323582.

Formula

The generating function for the numbers of espaliers of height h and volumes v_1 , ... v_h is 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.

Extensions

a(0)=1 prepended by Seiichi Manyama, Aug 20 2020