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A036366 Number of asymmetric n-ominoes in n-2 space. 1

%I

%S 0,1,4,13,42,113,309,792,2049,5167,13071,32724,82006,204619,510655,

%T 1272101,3168971,7888446,19636642,48868367,121621466,302673515,

%U 753319709,1875049668,4667676111,11620911254,28936281066,72062264255

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

%H W. F. Lunnon, <a href="https://doi.org/10.1093/comjnl/18.4.366">Counting Multidimensional Polyominoes</a>, Computer Journal, Vol. 18 (1975), pp. 366-67.

%F 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).

%e 0 asymmetric trominoes in 1-space;

%e 1 asymmetric tetromino in 2-space;

%e 4 asymmetric pentominoes in 3-space.

%t sa[ n_, k_ ] := sa[ n, k ]=a[ n+1-k, 1 ]+If[ n<2k, 0, -sa[ n-k, k ] ]; a[ 1, 1 ] := 1;

%t a[ n_, 1 ] := a[ n, 1 ]=Sum[ a[ i, 1 ]sa[ n-1, i ]i, {i, 1, n-1} ]/(n-1);

%t a[ n_, k_ ] := a[ n, k ]=Sum[ a[ i, 1 ]a[ n-i, k-1 ], {i, 1, n-1} ];

%t 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

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

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

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

%Y Cf. A004111, A000220, A036365.

%K easy,nice,nonn

%O 3,3

%A _Robert A. Russell_

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Last modified July 27 04:32 EDT 2021. Contains 346305 sequences. (Running on oeis4.)