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A229914
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Number of pyramid polycubes of a given volume in dimension 3.
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4
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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
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OFFSET
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1,2
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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 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.
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LINKS
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FORMULA
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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.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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