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A230119 Numbers of quasi-pyramid polycubes of a given volume (number of atomic cells) 0

%I #9 Jan 19 2019 04:14:59

%S 3,9,23,47,91,169,291,494,815,1295,2043,3155,4775,7165,10599,15458,

%T 22455,32145,45659,64519,90247,125493,173515,238153,325423,442169,

%U 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 Numbers of quasi-pyramid polycubes of a given volume (number of atomic cells)

%C 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 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 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 calcPyr:=proc(i, j, k, l) option remember;

%p if (l<0) then 0

%p elif (i*j*k>l) then 0

%p elif k=1 then if (i*j=l) then 1 else 0; fi;

%p else s:=0; a:=0; b:=0;

%p while ((i+a)*j*(k-1)<=l-i*j) do

%p b:=0;

%p while ((i+a)*(j+b)*(k-1)<=l-i*j) do

%p s:=s+(a+1)*(b+1)*calcPyr(i+a, j+b, k-1, l-i*j);

%p b:=b+1;

%p od;

%p a:=a+1;

%p od;

%p s;

%p fi;

%p end;

%p countPyr:=proc(l)

%p s:=0;

%p for k to l do

%p i:=1:

%p while (i*k<=l) do

%p j:=1;

%p while (i*k*j<=l) do

%p s:=s+t^k*calcRecPyr(i, j, k, l);

%p j:=j+1;

%p od:

%p i:=i+1;

%p od;

%p od;

%p s;

%p end;

%p enum=[seq(countPyr(ii), ii=1..200)];

%p serie_quasiPyr:=proc(l) global enum;local ii;

%p map(coeff,enump,t^l);

%p select(x->x>0,%);

%p sum(t^(ii-1)*%[ii],ii=1..nops(%));

%p end;

%p serie_quasiPyr(100):

%p [1,seq(coeff(%,t^ii)-1,ii=1..50)];

%Y A229914, A227926

%K nonn

%O 1,1

%A _Matthieu Deneufchâtel_, Oct 10 2013

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)