cp's OEIS Frontend

A230119 Numbers of quasi-pyramid polycubes of a given volume (number of atomic cells).

%I A230119 #11 Jun 02 2025 08:41:19
%S A230119 3,9,23,47,91,169,291,494,815,1295,2043,3155,4775,7165,10599,15458,
%T A230119 22455,32145,45659,64519,90247,125493,173515,238153,325423,442169,
%U A230119 597575,804203,1077283,1436593,1908571,2525611,3329391,4373969,5726611,7472763,9721983,12608475,16305179,21027921,27043631,34689187,44385995,56652421,72137483,91645813,116167379,146932651,185456419,233594265
%N A230119 Numbers of quasi-pyramid polycubes of a given volume (number of atomic cells).
%C A230119 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.
%C A230119 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).
%C A230119 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).
%p A230119 calcPyr:=proc(i, j, k, l) option remember;
%p A230119 if (l<0) then 0
%p A230119 elif (i*j*k>l) then 0
%p A230119 elif k=1 then if (i*j=l) then 1 else 0; fi;
%p A230119 else s:=0; a:=0; b:=0;
%p A230119 while ((i+a)*j*(k-1)<=l-i*j) do
%p A230119 b:=0;
%p A230119 while ((i+a)*(j+b)*(k-1)<=l-i*j) do
%p A230119 s:=s+(a+1)*(b+1)*calcPyr(i+a, j+b, k-1, l-i*j);
%p A230119 b:=b+1;
%p A230119 od;
%p A230119 a:=a+1;
%p A230119 od;
%p A230119 s;
%p A230119 fi;
%p A230119 end;
%p A230119 countPyr:=proc(l)
%p A230119 s:=0;
%p A230119 for k to l do
%p A230119 i:=1:
%p A230119 while (i*k<=l) do
%p A230119 j:=1;
%p A230119 while (i*k*j<=l) do
%p A230119 s:=s+t^k*calcRecPyr(i, j, k, l);
%p A230119 j:=j+1;
%p A230119 od:
%p A230119 i:=i+1;
%p A230119 od;
%p A230119 od;
%p A230119 s;
%p A230119 end;
%p A230119 enum=[seq(countPyr(ii), ii=1..200)];
%p A230119 serie_quasiPyr:=proc(l) global enum;local ii;
%p A230119 map(coeff,enump,t^l);
%p A230119 select(x->x>0,%);
%p A230119 sum(t^(ii-1)*%[ii],ii=1..nops(%));
%p A230119 end;
%p A230119 serie_quasiPyr(100):
%p A230119 [1,seq(coeff(%,t^ii)-1,ii=1..50)];
%Y A230119 A229914, A227926
%K A230119 nonn
%O A230119 1,1
%A A230119 _Matthieu Deneufchâtel_, Oct 10 2013