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 A230119 Numbers of quasi-pyramid polycubes of a given volume (number of atomic cells) 0
 3, 9, 23, 47, 91, 169, 291, 494, 815, 1295, 2043, 3155, 4775, 7165, 10599, 15458, 22455, 32145, 45659, 64519, 90247, 125493, 173515, 238153, 325423, 442169, 597575, 804203, 1077283, 1436593, 1908571, 2525611, 3329391, 4373969, 5726611, 7472763, 9721983, 12608475, 16305179, 21027921, 27043631, 34689187, 44385995, 56652421, 72137483, 91645813, 116167379, 146932651, 185456419, 233594265 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 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. A quasi-pyramid polycube is an object obtained from a pyramid by removing all the cells with coordinates (a,b,c) with 1 <= a <= h for a fixed pair (b,c) chosen among the triples (h,b,c) such that there is a cell with coordinates (h,b,c) in the pyramid ((h,b,c) belongs to the highest plateau of the pyramid). If Q(x) denotes the generating function of the quasi-pyramids and P(x,h) the generating function of the pyramids counted by height, then the x^(-h) P(x,h) converges when h goes to infinity and the limit is Q(x) + x/(1-x). LINKS MAPLE calcPyr:=proc(i, j, k, l) option remember; if (l<0) then 0 elif (i*j*k>l) then 0 elif k=1 then if (i*j=l) then 1 else 0; fi; else s:=0; a:=0; b:=0; while ((i+a)*j*(k-1)<=l-i*j) do b:=0; while ((i+a)*(j+b)*(k-1)<=l-i*j) do s:=s+(a+1)*(b+1)*calcPyr(i+a, j+b, k-1, l-i*j); b:=b+1; od; a:=a+1; od; s; fi; end; countPyr:=proc(l) s:=0; for k to l do i:=1: while (i*k<=l) do j:=1; while (i*k*j<=l) do s:=s+t^k*calcRecPyr(i, j, k, l); j:=j+1; od: i:=i+1; od; od; s; end; enum=[seq(countPyr(ii), ii=1..200)]; serie_quasiPyr:=proc(l) global enum; local ii; map(coeff, enump, t^l); select(x->x>0, %); sum(t^(ii-1)*%[ii], ii=1..nops(%)); end; serie_quasiPyr(100): [1, seq(coeff(%, t^ii)-1, ii=1..50)]; CROSSREFS Sequence in context: A210027 A018555 A147271 * A146693 A227259 A064551 Adjacent sequences: A230116 A230117 A230118 * A230120 A230121 A230122 KEYWORD nonn AUTHOR Matthieu Deneufchâtel, Oct 10 2013 STATUS approved

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Last modified January 31 21:41 EST 2023. Contains 359981 sequences. (Running on oeis4.)