A227926 Triangle read by rows: number of pyramid polycubes counted by height and volume.

1, 2, 1, 2, 4, 1, 3, 8, 4, 1, 2, 16, 10, 4, 1, 4, 22, 22, 10, 4, 1, 2, 36, 40, 24, 10, 4, 1, 4, 47, 66, 46, 24, 10, 4, 1, 3, 60, 110, 84, 48, 24, 10, 4, 1, 4, 83, 158, 144, 90, 48, 24, 10, 4, 1, 2, 100, 233, 232, 162, 92, 48, 24, 10, 4, 1, 6, 116, 327, 357, 266, 168, 92, 48, 24, 10, 4, 1, 2, 148, 444, 544, 435, 284, 170, 92, 48, 24, 10, 4, 1

Offset: 1

Author

Matthieu Deneufchâtel, Oct 09 2013

Keywords

• nonn
• tabl

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.

Links

• David Goodger, Polycubes: Puzzles & Solutions

Crossrefs

The numbers of pyramids counted by volume are given by A229914.

Programs

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

Formula

The number n_{i,j,h,v} of pyramids of volume v, height h and such that the highest plateau has volume i * j is given by the recurrence: n_{i,j,h,v} = Sum_{a=0..(i*j*h-v)/((h-1)*j)} Sum_{b=0..(j*(h*(i+a)-a)-v)/((i+a)*(k-1))} (a+1)*(b+1)*n_{i+a,j+a,h-1,v-i*j}.