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

1,2

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

Table of n, a(n) for n=1..91.

David Goodger, Polycubes: Puzzles & Solutions

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}.

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)];

Crossrefs

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

Sequence in context: A209750 A156042 A287691 * A357121 A348296 A357120

Adjacent sequences: A227923 A227924 A227925 * A227927 A227928 A227929

Keyword

nonn,tabl

Author

Matthieu Deneufchâtel, Oct 09 2013

Status

approved