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A229914 Number of pyramid polycubes of a given volume in dimension 3. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

REFERENCES

C. Carré, N. Debroux, M. Deneufchatel, J.-P. Dubernard et al., Dirichlet convolution and enumeration of pyramid polycubes, 2013; http://hal.archives-ouvertes.fr/docs/00/90/58/89/PDF/polycubes.pdf

LINKS

Table of n, a(n) for n=1..43.

C. Carre, N. Debroux, M. Deneufchatel, J.-Ph. Dubernard, C. Hillariet, J.-G. Luque, O. Mallet, Enumeration of Polycubes and Dirichlet Convolutions, J. Int. Seq. 18 (2015) 15.11.4

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.

CROSSREFS

A001931 is an upper bound.

Sequence in context: A161810 A084631 A219846 * A192968 A277968 A217942

Adjacent sequences:  A229911 A229912 A229913 * A229915 A229916 A229917

KEYWORD

nonn

AUTHOR

Matthieu Deneufchâtel, Oct 03 2013

STATUS

approved

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Last modified June 25 15:30 EDT 2018. Contains 311914 sequences. (Running on oeis4.)