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 1-3 of 3 results.

A355048 Number of unoriented orthoplex n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

3, 18, 122, 655, 3240, 14531, 61520, 247381, 958434, 3598594, 13180348, 47274577, 166642096, 578750970, 1984671466, 6731351834, 22612409886, 75321920403, 249028297179, 817867225710, 2670093233760, 8670380548402
Offset: 6

Views

Author

Robert A. Russell, Jun 16 2022

Keywords

Comments

Orthoplex polyominoes are connected sets of cells of regular tilings with Schläfli symbols {}, {4}, {3,4}, {3,3,4}, {3,3,3,4}, etc. These are tilings of regular orthoplexes projected on their circumspheres. Orthoplex polyominoes are equivalent to multidimensional polyominoes that do not extend more than two units along any axis, i.e., fit within a 2^d cube. For unoriented polyominoes, chiral pairs are counted as one.

Examples

			a(6)=3 because there are 3 hexominoes in 2^3 space. The two vacant cells share just a face, an edge, or a vertex.

Links

Crossrefs

Cf. A355047 (oriented), A355049 (chiral), A355050 (achiral) A355051 (asymmetric), A000081 (rooted trees).
Other dimensions: A036367 (n-2), A000055 (n-1), A355053 (multidimensional).

Programs

  • Mathematica
    sb[n_,k_] := sb[n,k] = b[n+1-k,1] + If[n<2k,0,sb[n-k,k]];
b[1,1] := 1; b[n_,1] := b[n,1] = Sum[b[i,1]sb[n-1,i]i,{i,1,n-1}]/(n-1);
b[n_,k_] := b[n,k] = Sum[b[i,1]b[n-i,k-1],{i,1,n-1}];
nmax = 30; B[x_] := Sum[b[i,1]x^i,{i,0,nmax}]
Drop[CoefficientList[Series[(14B[x]^6 + 3B[x]^7 + 6B[x]^4B[x^2] + 6B[x]^5B[x^2] + 18B[x]^2B[x^2]^2 + 3B[x]^3B[x^2]^2 + 26B[x^2]^3 + 6 B[x]B[x^2](B[x^2]^2 + B[x^4]) + 4B[x^3]^2 + 4B[x^6]) / 24 + B[x]^3 (38B[x]^4 + 9B[x]^5 + 4B[x]^2B[x^2] + 10B[x]^3B[x^2] + 2B[x^2]^2 + B[x]B[x^2]^2) / (8(1-B[x])) + B[x]^6 (16B[x]^2 + 6B[x]^3 + B[x^2] + B[x] (5 + 2B[x^2])) / (2(1-B[x])^2) + B[x]^7 (2 + 42B[x] + 51B[x]^2 + 24B[x]^3 + 3B[x^2]) / (12(1-B[x])^3) + B[x]^9 (17 + 8B[x]) / (8(1-B[x])^4) + 3B[x]^10 / (8(1-B[x])^5) + B[x^2]^2(B[x]^4 + 4B[x]^2 B[x^2] + 12B[x^2]^2 + B[x^4] + B[x] (8B[x^2] + 5B[x^2]^2 + B[x^4])) / (4(1-B[x^2])) + B[x^2]^4 (8 + 16B[x^2] + B[x] (19 + 8B[x^2])) / (8(1-B[x^2])^2) + 3(1 + B[x])B[x^2]^5 / (4(1-B[x^2])^3) + 2B[x]B[x^3]^2 / (6(1-B[x^3])) + B[x]B[x^4]^2 / (4(1-B[x^4])) + B[x]^2B[x^2]^2(5B[x]^3 + 2B[x^2] + B[x](2 + B[x^2])) / (4(1-B[x])(1-B[x^2])) + B[x]^5(1+4B[x])B[x^2]^2 / (4(1-B[x])^2(1-B[x^2])) + B[x]^6 B[x^2]^2 / (4(1-B[x])^3(1-B[x^2])) + 3B[x]^2B[x^2]^4 / (8(1-B[x])(1-B[x^2])^2) + B[x^2](1+B[x])B[x^4]^2 / (4(1-B[x^2])(1-B[x^4])), {x,0,nmax}],x],6]

Formula

a(n) = A355047(n) - A355049(n) = (A355047(n) + A355050(n)) / 2 = A355049(n) + A355050(n).
G.f.: (14*B(x)^6 + 3*B(x)^7 + 6*B(x)^4*B(x^2) + 6*B(x)^5*B(x^2) + 18*B(x)^2*B(x^2)^2 + 3*B(x)^3*B(x^2)^2 + 26*B(x^2)^3 + 6*B(x)*B(x^2)*(B(x^2)^2 + B(x^4)) + 4*B(x^3)^2 + 4*B(x^6)) / 24 + B(x)^3*(38*B(x)^4 + 9*B(x)^5 + 4*B(x)^2*B(x^2) + 10*B(x)^3*B(x^2) + 2*B(x^2)^2 + B(x)*B(x^2)^2) / (8*(1-B(x))) + B(x)^6*(16*B(x)^2 + 6*B(x)^3 + B(x^2) + B(x)*(5 + 2*B(x^2))) / (2*(1-B(x))^2) + B(x)^7*(2 + 42*B(x) + 51*B(x)^2 + 24*B(x)^3 + 3*B(x^2)) / (12*(1-B(x))^3) + B(x)^9*(17 + 8*B(x)) / (8*(1-B(x))^4) + 3*B(x)^10 / (8*(1-B(x))^5) + B(x^2)^2*(B(x)^4 + 4*B(x)^2*B(x^2) + 12*B(x^2)^2 + B(x^4) + B(x)*(8*B(x^2) + 5*B(x^2)^2 + B(x^4))) / (4*(1-B(x^2))) + B(x^2)^4*(8 + 16*B(x^2) + B(x)*(19 + 8*B(x^2))) / (8*(1-B(x^2))^2) + 3*(1 + B(x))*B(x^2)^5 / (4*(1-B(x^2))^3) + 2*B(x)*B(x^3)^2 / (6*(1-B(x^3))) + B(x)*B(x^4)^2 / (4*(1-B(x^4))) + B(x)^2*B(x^2)^2*(5*B(x)^3 + 2*B(x^2) + B(x)*(2 + B(x^2))) / (4*(1-B(x))*(1-B(x^2))) + B(x)^5*(1+4*B(x))*B(x^2)^2 / (4*(1-B(x))^2*(1-B(x^2))) + B(x)^6*B(x^2)^2 / (4*(1-B(x))^3*(1-B(x^2))) + 3*B(x)^2*B(x^2)^4 / (8*(1-B(x))*(1-B(x^2))^2) + B(x^2)*(1+B(x))*B(x^4)^2 / (4*(1-B(x^2))*(1-B(x^4))), where B(x) is the generating function for rooted trees with n nodes in A000081.

A036368 Number of chiral orthoplex n-ominoes in n-2 space.

Original entry on oeis.org

0, 0, 4, 14, 37, 110, 324, 888, 2368, 6336, 16874, 44414, 116181, 303362, 790157, 2051880, 5317599, 13764133, 35586766, 91910082, 237183164, 611701614, 1576773162, 4062606255, 10463699696, 26942811809, 69358469092
Offset: 4

Views

Author

Robert A. Russell

Keywords

Comments

Orthoplex polyominoes are multidimensional polyominoes that do not extend more than two units along any axis.

Examples

			a(6)=4 because there are 4 pairs of chiral hexominoes in 2^4 space.

Crossrefs

Cf. A045648, A036367.

Programs

  • Mathematica
    sc[ n_, k_ ] := sc[ n, k ]=c[ n+1-k, 1 ]+If[ n<2k, 0, sc[ n-k, k ](-1)^k ]; c[ 1, 1 ] := 1;
c[ n_, 1 ] := c[ n, 1 ]=Sum[ c[ i, 1 ]sc[ n-1, i ]i, {i, 1, n-1} ]/(n-1);
c[ n_, k_ ] := c[ n, k ]=Sum[ c[ i, 1 ]c[ n-i, k-1 ], {i, 1, n-1} ];
Table[ c[ i, 4 ]/8+Sum[ c[ i, j ], {j, 5, i} ]/2-If[ OddQ[ i ], 0,
c[ i/2, 2 ](-1)^(i/2)/8+If[ OddQ[ i/2 ], 0, c[ i/4, 1 ](-1)^(i/4)/4 ]
+Sum[ c[ i/2, j ](-1)^(i/2), {j, 3, i/2} ]/2 ]+Sum[ c[ j, 1 ]c[ i-2j, 2 ](-1)^j/4
-Sum[ If[ OddQ[ k ], c[ j, (k-1)/2 ]c[ i-2j, 1 ](-1)^j/2, 0 ], {k, 5, i} ],
{j, 1, (i-1)/2} ], {i, 4, 30} ]

Formula

G.f.: (C^2(x) + C(-x^2))^2/8 - C^2(-x^2)/4 - C(-x^4)/4 + C^5(x)/(2-2C(x)) - (C(x)+C(-x^2))*C^2(-x^2)/(2-2C(-x^2)) where C(x) is the generating function for chiral n-ominoes in n-1 space, one cell labeled in A045648.

A355998 Number of fixed orthoplex n-ominoes with cell centers determining (n-2)-space.

Original entry on oeis.org

1, 48, 1728, 62720, 2457600, 105815808, 5017600000, 261227298816, 14860167413760, 918839084134400, 61439672177393700, 4421589120000000000, 340976534987475000000, 28064307240230900000000, 2456376885785930000000000
Offset: 4

Views

Author

Robert A. Russell, Jul 22 2022

Keywords

Comments

Orthoplex polyominoes are connected sets of cells of regular tilings with Schläfli symbols {}, {4}, {3,4}, {3,3,4}, {3,3,3,4}, etc. These are tilings of regular orthoplexes projected on their circumspheres. Orthoplex polyominoes are equivalent to multidimensional polyominoes that do not extend more than two units along any axis, i.e., fit within a 2^d cube. Two fixed polyominoes are identical only if one is a translation of the other.

Examples

			For A(4)=1, all 4 squares of the 2^2 space are used.

Crossrefs

Cf. A171860 (multidimensional), A036367 (unoriented), A036368 (chiral), A036369 (asymmetric).
Diagonal 2 of A355997.

Programs

  • Mathematica
    Table[2^(n-3) n^(n-5) (n-2) (n-3)^2, {n,4,30}]

Formula

a(n) = 2^(n-3) * n^(n-5) * (n-2) * (n-3)^2.
a(n) ~ A171860(n) / 2.
Showing 1-3 of 3 results.

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