3,3

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

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^2(x^2)/(1-A(x^2)) where A(x) is generating function for rooted identity trees with n nodes

0 asymmetric trominoes in 1-space; 1 asymmetric tetromino in 2-space;

4 asymmetric pentominoes in 3-space

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} ]

Cf. A004111, A000220, A036365.

Sequence in context: A005002 A085507 A121654 this_sequence A109454 A000640 A010919

Adjacent sequences: A036363 A036364 A036365 this_sequence A036367 A036368 A036369

easy,nice,nonn

Robert A. Russell (russell(AT)post.harvard.edu)