cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Matthieu Deneufchâtel, Oct 09 2013

Keywords

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

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}.

A229916 Numbers of pyramid polycubes of a given volume in dimension 4.

Original entry on oeis.org

1, 4, 10, 25, 55, 109, 211, 371, 651, 1092, 1767, 2775, 4383, 6666, 9990, 14806, 21695, 31265, 44852, 63365, 89132, 124250, 171560, 235140, 321661, 435670, 587219, 787663, 1051669, 1396309, 1848190, 2432288, 3192615, 4174278, 5435945, 7054030, 9134731, 11779666, 15152584, 19436993, 24867486
Offset: 1

Views

Author

Matthieu Deneufchâtel, Oct 03 2013

Keywords

Comments

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 with coordinates (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.

Crossrefs

Cf. A229914, A227926.

A229924 Numbers of pyramid polycubes of a given volume in dimension 5.

Original entry on oeis.org

1, 5, 13, 35, 81, 165, 329, 587, 1059, 1807, 2939, 4633, 7431, 11391, 17091, 25372, 37343, 53913, 77393, 109273, 153857, 214409, 295293, 403545, 551715, 745643, 1001721, 1339677, 1784109
Offset: 1

Views

Author

Matthieu Deneufchâtel, Oct 03 2013

Keywords

Comments

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.

Crossrefs

Cf. A229914, A227926.

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

Original entry on oeis.org

3, 9, 23, 47, 91, 169, 291, 494, 815, 1295, 2043, 3155, 4775, 7165, 10599, 15458, 22455, 32145, 45659, 64519, 90247, 125493, 173515, 238153, 325423, 442169, 597575, 804203, 1077283, 1436593, 1908571, 2525611, 3329391, 4373969, 5726611, 7472763, 9721983, 12608475, 16305179, 21027921, 27043631, 34689187, 44385995, 56652421, 72137483, 91645813, 116167379, 146932651, 185456419, 233594265
Offset: 1

Views

Author

Matthieu Deneufchâtel, Oct 10 2013

Keywords

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.
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).
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).

Crossrefs

A229914, A227926

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;
enum=[seq(countPyr(ii), ii=1..200)];
serie_quasiPyr:=proc(l) global enum;local ii;
map(coeff,enump,t^l);
select(x->x>0,%);
sum(t^(ii-1)*%[ii],ii=1..nops(%));
end;
serie_quasiPyr(100):
[1,seq(coeff(%,t^ii)-1,ii=1..50)];
Showing 1-4 of 4 results.

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