A036366 - OEIS

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A036366

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

0, 1, 4, 13, 42, 113, 309, 792, 2049, 5167, 13071, 32724, 82006, 204619, 510655, 1272101, 3168971, 7888446, 19636642, 48868367, 121621466, 302673515, 753319709, 1875049668, 4667676111, 11620911254, 28936281066, 72062264255

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OFFSET

3,3

LINKS

Table of n, a(n) for n=3..30.

W. F. Lunnon, Counting Multidimensional Polyominoes, Computer Journal, Vol. 18 (1975), pp. 366-67.

FORMULA

G.f.: A^3(x)/2 - A(x)A(x^2)/2 + 5A^4(x)/8 - A^2(x)A(x^2)/4 - 5A^2(x^2)/8 + A(x^4)/4 + A^5(x)/(1-A(x)) - (A(x)+A(x^2))*A^2(x^2)/(1-A(x^2)), where A(x) is the generating function for rooted identity trees with n nodes (that is, the g.f. of sequence A004111).

EXAMPLE

0 asymmetric trominoes in 1-space;

1 asymmetric tetromino in 2-space;

4 asymmetric pentominoes in 3-space.

MATHEMATICA

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, 3 ]/2+5a[ i, 4 ]/8+Sum[ a[ i, j ], {j, 5, i} ]-If[ OddQ[ i ], 0, 5a[ i/2, 2 ]/8

-If[ OddQ[ i/2 ], 0, a[ i/4, 1 ]/4 ]+Sum[ a[ i/2, j ], {j, 3, i/2} ] ]

-Sum[ a[ j, 1 ](a[ i-2j, 1 ]/2+a[ i-2j, 2 ]/4)+Sum[ If[ OddQ[ k ], a[ j,

(k-1)/2 ]a[ i-2j, 1 ], 0 ], {k, 5, i} ], {j, 1, (i-1)/2} ], {i, 3, 30} ]