A227925 -id:A227925 - cp's OEIS frontend

A229915 Number of espalier polycubes of a given volume in dimension 3.

1, 1, 3, 5, 10, 14, 26, 34, 57, 76, 116, 150, 227, 284, 408, 520, 718, 895, 1226, 1508, 2018, 2487, 3248, 3968, 5160, 6235, 7970, 9653, 12179, 14630, 18367, 21924, 27241, 32506, 39985, 47492, 58203, 68752, 83613, 98730, 119269, 140224, 168799, 197758, 236753, 277052, 329867, 384852, 457006, 531500, 628338

Offset: 0

Matthieu Deneufchâtel, Oct 03 2013

nonn

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 coordinates (0,b,c) belongs to the first plateau 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 coordinates (a,0,0).

Seiichi Manyama, Table of n, a(n) for n = 0..100
C. Carré, N. Debroux, M. Deneufchatel, J.-P. Dubernard et al., Dirichlet convolution and enumeration of pyramid polycubes, 2013.
C. Carre, N. Debroux, M. Deneufchatel, J.-Ph. Dubernard, C. Hillariet, J.-G. Luque, and O. Mallet, Enumeration of Polycubes and Dirichlet Convolutions, J. Int. Seq. 18 (2015) 15.11.4.

Cf. A026820, A100882, A227925, A230118, A229917, A229925, A323582.

The generating function for the numbers of espaliers of height h and volumes v_1 , ... v_h is x_1^{n_1} * ... x_h^{n_h} / ((1-x_1^{n_1}) *(1-x_1^{n_1}*x_2^{n_2}) *... *(1-x_1^{n_1}*x_2^{n_2}*...x_h^{n_h})).

This sequence is obtained with x_1 = ... = x_h = p by summing over n_1>= ... >= n_h>=1 and then over h.

a(0)=1 prepended by Seiichi Manyama, Aug 20 2020

A229917 Numbers of espalier polycubes of a given volume in dimension 4.

Original entry on oeis.org

1, 4, 7, 16, 22, 46, 58, 107, 140, 227, 287, 464, 563, 851, 1067, 1530, 1866, 2661, 3198, 4428, 5361, 7185, 8613, 11524, 13639, 17839, 21272, 27359, 32300, 41369, 48512, 61311, 72105, 89904, 105226, 130834, 152164, 187297, 218356, 266444, 309125, 375995, 434670, 525045, 607329, 728256, 839874, 1004938

Offset: 1

Matthieu Deneufchâtel, Oct 03 2013

nonn

A (d+1)-pyramid polycube is a (d+1)-polycube obtained by gluing together horizontal (d+1)-plateaux (parallelepipeds of height 1) in such a way that the cell (0,0,...,0) belongs to the first plateau and each cell with coordinates (0,n_1,...,n_d) belonging to the first plateau is such that n_1 , ... , n_d >= 0.
If the cell with coordinates (n_0,n_1,...,n_d) belongs to the (n_0+1)-st plateau (n_0>0), then the cell with coordinates (n_0-1, n_1, ... ,n_d) belongs to the n_0-th plateau.
A (d+1)-espalier is a (d+1)-pyramid such that each plateau contains the cell (n_0,0,...,0).

Cf. A229915, A227925.

A229925 Numbers of espalier polycubes of a given volume in dimension 5.

Original entry on oeis.org

1, 5, 9, 23, 31, 71, 87, 173, 223, 379, 471, 801, 951, 1495, 1851, 2736, 3282, 4832, 5708, 8126, 9704, 13290, 15694, 21496, 25038, 33396, 39330, 51452

Offset: 1

Matthieu Deneufchâtel, Oct 03 2013

nonn

A (d+1)-pyramid polycube is a (d+1)-polycube obtained by gluing together horizontal (d+1)-plateaux (parallelepipeds of height 1) in such a way that the cell (0,0,...,0) belongs to the first plateau and each cell with coordinates (0,n_1,...,n_d) belonging to the first plateau is such that n_1 , ... , n_d >= 0.
If the cell with coordinates (n_0,n_1,...,n_d) belongs to the (n_0+1)-st plateau (n_0>0), then the cell with coordinates (n_0-1, n_1, ... ,n_d) belongs to the n_0-th plateau.
A (d+1)-espalier is a (d+1)-pyramid such that each plateau contains the cell (n_0,0,...,0).

Cf. A227925, A229915.

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

Original entry on oeis.org

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

Matthieu Deneufchâtel, Oct 10 2013

nonn

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.

Cf. A227925, A229915.

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)];

cp's OEIS Frontend

A229915 Number of espalier polycubes of a given volume in dimension 3.

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A229917 Numbers of espalier polycubes of a given volume in dimension 4.

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A229925 Numbers of espalier polycubes of a given volume in dimension 5.

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

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