A230118 Numbers of quasi-espalier polycubes of a given volume (number of atomic cells).

1, 2, 4, 7, 12, 18, 29, 42, 61, 87, 122, 167, 229, 306, 409, 538, 705, 915, 1182, 1509, 1927, 2438, 3075, 3854, 4814, 5985, 7416, 9144, 11253, 13784, 16845, 20512, 24922, 30179, 36470, 43939, 52841, 63378, 75864, 90605, 108022, 128496, 152603, 180865, 214044, 252826, 298192, 351108, 412832, 484632, 568157

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.

An espalier polycube is a special pyramid such that each plateau contains the cell with coordinate (a,0,0).

Quasi espaliers are espaliers from which all the cells with coordinates (a,0,0) have been removed.

If E(x,h) denotes the generating function of espalier polycubes, x^(-h) E(x,h) converges when h tends to infinity towards a series which is the generating function of quasi-espalier polycubes.

Links

Table of n, a(n) for n=1..51.

Maple

calcRecEsp:=proc(i, j, k, l) option remember;
## Compute the number n_{i, j, k, l}
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+calcRecEsp(i+a, j+b, k-1, l-i*j);
b:=b+1;
od;
a:=a+1;
od;
s;
fi;
end;
compteEsp:=proc(l) ### compute \sum_{v}n_{h, v}t^v
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*calcRecEsp(i, j, k, l);
j:=j+1;
od:
i:=i+1;
od;
od;
s;
end;
enum = [seq(op(convert(compteEsp(ii), list)), ii=2..200)];
serie_quasi_Esp:=proc(l)global enum; local ii;
map(coeff, enum, t^l);
select(x->x>0, %);
sum(t^(ii-1)*%[ii], ii=1..nops(%));
end;
serie_quasi_Esp(100):
[1, seq(coeff(%, t^ii)-1, ii=1..50)];

Crossrefs

Cf. A227925, A229915.

Sequence in context: A353150 A035300 A035296 * A105807 A209616 A192521

Adjacent sequences: A230115 A230116 A230117 * A230119 A230120 A230121

Keyword

nonn

Author

Matthieu Deneufchâtel, Oct 10 2013

Status

approved