OFFSET
4,3
COMMENTS
Orthoplex polyominoes are multidimensional polyominoes that do not extend more than two units along any axis.
FORMULA
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.
EXAMPLE
a(6)=4 because there are 2 asymmetric hexominoes in 2^4 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, 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} ]
CROSSREFS
KEYWORD
easy,nice,nonn
AUTHOR
STATUS
approved