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A036365 Number of chiral n-ominoes in n-2 space. 3
0, 2, 6, 17, 49, 135, 361, 951, 2493, 6497, 16837, 43498, 112164, 288741, 742294, 1906552, 4893835, 12555662, 32201344, 82566738, 211675672, 542621858, 1390929877, 3565435302, 9139718572, 23430209922, 60069035611, 154014868677 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

a(n) is Lunnon's DR(n,n-2) - DE(n,n-2).

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.: C^3(x)/2 + C(x)C(-x^2)/2 + 5C^4(x)/8 + C^2(x)C(-x^2)/4 + 3C^2(-x^2)/8 - C(-x^4)/4 + C^5(x)/(1-C(x)), where C(x) is the generating function for chiral n-ominoes in n-1 space, one cell labeled (that is, C(x) is the g.f. of the sequence A045648).

EXAMPLE

0 chiral trominoes in 1-space;

2 pairs of chiral tetrominoes (L,S) in 2-space;

6 pairs of chiral pentominoes in 3-space.

MATHEMATICA

sc[ n_, k_ ] := sc[ n, k ]=c[ n+1-k, 1 ]+If[ n<2k, 0, sc[ n-k, k ](-1)^k ]; c[ 1, 1 ] := 1;

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

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

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

3c[ i/2, 2 ](-1)^(i/2)/8-If[ OddQ[ i/2 ], 0, c[ i/4, 1 ](-1)^(i/4)/4 ] ]

+Sum[ c[ j, 1 ](c[ i-2j, 1 ]/2+c[ i-2j, 2 ]/4)(-1)^j, {j, 1, (i-1)/2} ], {i, 3, 30} ]

CROSSREFS

Cf. A045648, A045649, A036364.

Sequence in context: A019487 A077936 A077983 * A299162 A244400 A052536

Adjacent sequences:  A036362 A036363 A036364 * A036366 A036367 A036368

KEYWORD

easy,nice,nonn

AUTHOR

Robert A. Russell

STATUS

approved

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Last modified June 12 08:13 EDT 2021. Contains 344943 sequences. (Running on oeis4.)