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A036367 Number of free orthoplex n-ominoes with cell centers determining n-2 space. 1
1, 2, 8, 25, 86, 272, 875, 2732, 8505, 26104, 79708, 241522, 728632, 2187951, 6548819, 19542662, 58184124, 172880565, 512837063, 1519158462, 4494920802, 13286473612, 39240530012, 115811180864, 341588823740, 1007007175952, 2967361180383 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,2

COMMENTS

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

LINKS

Table of n, a(n) for n=4..30.

FORMULA

G.f.: (B^2(x) + B(x^2))^2/8 + B^2(x^2)/4 + B(x^4)/4 + B^5(x)/(2 - 2*B(x)) + (B(x) + B(x^2))*B^2(x^2)/(2 - 2*B(x^2)) where B(x) is the generating function for rooted trees with n nodes in A000081.

EXAMPLE

a(4)=1 because there is 1 tetromino (a square) in 2^2 space;

a(5)=2 because there are 2 pentominoes in 2^3 space;

a(6)=8 because in 2^4 space there are 8 hexominoes that have cell centers determining 4-space.

MATHEMATICA

sb[ n_, k_ ] := sb[ n, k ]=b[ n+1-k, 1 ]+If[ n<2k, 0, sb[ n-k, k ] ]; b[ 1, 1 ] := 1;

b[ n_, 1 ] := b[ n, 1 ]=Sum[ b[ i, 1 ]sb[ n-1, i ]i, {i, 1, n-1} ]/(n-1);

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

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

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

+Sum[ b[ j, 1 ]b[ i-2j, 2 ]/4+Sum[ If[ OddQ[ k ], b[ j,

(k-1)/2 ]b[ i-2j, 1 ], 0 ], {k, 5, i} ]/2, {j, 1, (i-1)/2} ], {i, 4, 30} ]

CROSSREFS

Cf. A000081, A036364.

Sequence in context: A240478 A288539 A281338 * A115256 A132963 A122404

Adjacent sequences:  A036364 A036365 A036366 * A036368 A036369 A036370

KEYWORD

easy,nice,nonn

AUTHOR

Robert A. Russell

STATUS

approved

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Last modified June 18 18:09 EDT 2021. Contains 345120 sequences. (Running on oeis4.)