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A229915
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Number of espalier polycubes of a given volume in dimension 3.
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6
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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
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OFFSET
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0,3
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COMMENTS
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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).
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LINKS
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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.
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FORMULA
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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.
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CROSSREFS
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Cf. A026820, A100882, A227925, A230118, A229917, A229925, A323582.
Sequence in context: A320886 A323433 A220489 * A092269 A323429 A319066
Adjacent sequences: A229912 A229913 A229914 * A229916 A229917 A229918
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KEYWORD
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nonn
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AUTHOR
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Matthieu Deneufchâtel, Oct 03 2013
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EXTENSIONS
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a(0)=1 prepended by Seiichi Manyama, Aug 20 2020
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STATUS
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approved
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