A036366 -id:A036366 - cp's OEIS frontend

A355056 Number of asymmetric multidimensional n-ominoes with cell centers determining n-3 space.

5, 46, 275, 1283, 5281, 19607, 68476, 227196, 727780, 2263148, 6881482, 20529511, 60312548, 174870492, 501443277, 1424142358, 4011274417, 11216074419, 31160837273, 86078096135, 236568911194, 647181951619

Offset: 5

Robert A. Russell, Jun 16 2022

Multidimensional polyominoes are connected sets of cells of regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). An asymmetric polyomino has a symmetry group of order 1.

			a(5)=5 as there are exactly five asymmetric pentominoes in 2-space.

Robert A. Russell, Table of n, a(n) for n = 5..100
Robert A. Russell, Trunk Generating Functions

Cf. A355052 (oriented), A355053 (unoriented), A355054 (chiral), A355055 (achiral), A191092 (fixed), A004111 (rooted asymmetric).

Other dimensions: A036366 (n-2), A000220 (n-1), A355051 (orthoplex).

sa[n_, k_] := sa[n, k] = a[n+1-k, 1] + If[n < 2 k, 0, -sa[n-k, k]];
a[1, 1] := 1; a[n_, 1] := a[n, 1] = Sum[a[i, 1] sa[n-1, i] i, {i, 1, n-1}]/(n-1);
a[n_, k_] := a[n, k] = Sum[a[i, 1] a[n-i, k-1], {i, 1, n-1}];
nmax = 30; A[x_] := Sum[a[i, 1] x^i, {i, 0, nmax}]
Drop[CoefficientList[Series[(4 A[x]^4 + 37 A[x]^5 + 12 A[x]^6 - 6 A[x]^3 A[x^2] - 10 A[x]^4 A[x^2] - 4 A[x^2]^2 - 17 A[x] A[x^2]^2 - 2 A[x^2]^3 + 2 A[x] A[x^4]) / 8 + (24 A[x]^5 + 515 A[x]^6 + 325 A[x]^7 + 24 A[x]^8 - 48 A[x]^4 A[x^2] - 96 A[x]^5 A[x^2] - 24 A[x]^6 A[x^2] - 21 A[x]^2 A[x^2]^2 + 21 A[x]^3 A[x^2]^2 - 14 A[x^2]^3 + 8 A[x] A[x^2]^3 + 6 A[x]^2 A[x^2]^3 + 4 A[x^3]^2 - 4 A[x] A[x^3]^2 + 24 A[x^2] A[x^4] - 18 A[x] A[x^2] A[x^4] - 6 A[x]^2 A[x^2] A[x^4] - 4 A[x^6] + 4 A[x] A[x^6]) / (24 (1-A[x])) + A[x]^5 (2 A[x] + 67 A[x]^2 + 46 A[x]^3 + 6 A[x]^4 - 3 A[x^2] - 6 A[x] A[x^2] - 2 A[x]^2 A[x^2]) / (2 (1-A[x])^2) - A[x^2] (2 A[x]^2 A[x^2] + 6 A[x]^3 A[x^2] + 2 A[x]^4 A[x^2] + 13 A[x^2]^2 + 31 A[x] A[x^2]^2 + 2 A[x]^2 A[x^2]^2 + 15 A[x^2]^3 + 5 A[x] A[x^2]^3 - 3 A[x^4] - 5 A[x] A[x^4] - 3 A[x^2] A[x^4] - A[x] A[x^2] A[x^4]) / (4 (1-A[x^2])) + A[x]^6 (4 A[x] + 153 A[x]^2 + 75 A[x]^3 + 12 A[x]^4 - 3 A[x^2] - 3 A[x] A[x^2]) / (6 (1-A[x])^3) - A[x]^2 A[x^2]^2 (2 A[x] + 7 A[x]^2 + 5 A[x]^3 + A[x^2] - A[x] A[x^2]) / (2 (1-A[x]) (1-A[x^2])) + A[x] A[x^3]^2 / (1-A[x^3]) / 3 + A[x]^9 (21 + 4 A[x]) / (2 (1-A[x])^4) - A[x]^5 (3 + 2 A[x]) A[x^2]^2 / ((1-A[x])^2 (1-A[x^2])) - A[x^2]^4 (5 + 7 A[x] + 3 A[x^2] + A[x] A[x^2]) / (1-A[x^2])^2 + A[x] A[x^4]^2 / (2 (1-A[x^4])) + 3 A[x]^10 / (2 (1-A[x])^5) - A[x]^6 A[x^2]^2 / ((1-A[x])^3 (1-A[x^2])) - 2 (1 + A[x]) A[x^2]^5 / (1-A[x^2])^3 + 3 (1 + A[x]) A[x^2] A[x^4]^2 / (2 (1-A[x^2]) (1-A[x^4])), {x,0,nmax}], x], 5]

G.f.: (4 A(x)^4 + 37 A(x)^5 + 12 A(x)^6 - 6 A(x)^3 A(x^2) - 10 A(x)^4 A(x^2) - 4 A(x^2)^2 - 17 A(x) A(x^2)^2 - 2 A(x^2)^3 + 2 A(x) A(x^4)) / 8 + (24 A(x)^5 + 515 A(x)^6 + 325 A(x)^7 + 24 A(x)^8 - 48 A(x)^4 A(x^2) - 96 A(x)^5 A(x^2) - 24 A(x)^6 A(x^2) - 21 A(x)^2 A(x^2)^2 + 21 A(x)^3 A(x^2)^2 - 14 A(x^2)^3 + 8 A(x) A(x^2)^3 + 6 A(x)^2 A(x^2)^3 + 4 A(x^3)^2 - 4 A(x) A(x^3)^2 + 24 A(x^2) A(x^4) - 18 A(x) A(x^2) A(x^4) - 6 A(x)^2 A(x^2) A(x^4) - 4 A(x^6) + 4 A(x) A(x^6)) / (24 (1-A(x))) + A(x)^5 (2 A(x) + 67 A(x)^2 + 46 A(x)^3 + 6 A(x)^4 - 3 A(x^2) - 6 A(x) A(x^2) - 2 A(x)^2 A(x^2)) / (2 (1-A(x))^2) - A(x^2) (2 A(x)^2 A(x^2) + 6 A(x)^3 A(x^2) + 2 A(x)^4 A(x^2) + 13 A(x^2)^2 + 31 A(x) A(x^2)^2 + 2 A(x)^2 A(x^2)^2 + 15 A(x^2)^3 + 5 A(x) A(x^2)^3 - 3 A(x^4) - 5 A(x) A(x^4) - 3 A(x^2) A(x^4) - A(x) A(x^2) A(x^4)) / (4 (1-A(x^2))) + A(x)^6 (4 A(x) + 153 A(x)^2 + 75 A(x)^3 + 12 A(x)^4 - 3 A(x^2) - 3 A(x) A(x^2)) / (6 (1-A(x))^3) - A(x)^2 A(x^2)^2 (2 A(x) + 7 A(x)^2 + 5 A(x)^3 + A(x^2) - A(x) A(x^2)) / (2 (1-A(x)) (1-A(x^2))) + A(x) A(x^3)^2 / (1-A(x^3)) / 3 + A(x)^9 (21 + 4 A(x)) / (2 (1-A(x))^4) - A(x)^5 (3 + 2 A(x)) A(x^2)^2 / ((1-A(x))^2 (1-A(x^2))) - A(x^2)^4 (5 + 7 A(x) + 3 A(x^2) + A(x) A(x^2)) / (1-A(x^2))^2 + A(x) A(x^4)^2 / (2 (1-A(x^4))) + 3 A(x)^10 / (2 (1-A(x))^5) - A(x)^6 A(x^2)^2 / ((1-A(x))^3 (1-A(x^2))) - 2 (1 + A(x)) A(x^2)^5 / (1-A(x^2))^3 + 3 (1 + A(x)) A(x^2) A(x^4)^2 / (2 (1-A(x^2)) (1-A(x^4))) where A(x) is the generating function for rooted identity trees with n nodes in A004111.

A036369 Number of asymmetric orthoplex n-ominoes in n-2 space.

Original entry on oeis.org

0, 0, 2, 10, 33, 101, 277, 754, 1972, 5134, 13145, 33563, 85048, 215075, 541887, 1363376, 3423537, 8588521, 21522243, 53898911, 134895022, 337467395, 843924068, 2109897826, 5273789335, 13180026621, 32934865984

Offset: 4

Robert A. Russell

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

			a(6)=4 because there are 2 asymmetric hexominoes in 2^4 space.

Cf. A004111, A036366.

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

G.f.: (A^2(x)-A(x^2))^2/8 - A^2(x^2)/4 + A(x^4)/4 + A^5(x)/(2-2A(x)) - (A(x)+A(x^2))*A^2(x^2)/(2-2A(x^2)) where A(x) is the generating function for rooted identity trees with n nodes in A004111.

cp's OEIS Frontend

A355056 Number of asymmetric multidimensional n-ominoes with cell centers determining n-3 space.

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