A036365 Number of chiral n-ominoes in n-2 space.

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

Offset: 3

Author

Robert A. Russell

Keywords

• easy
• nice
• nonn

Comments

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

Examples

			0 chiral trominoes in 1-space;
2 pairs of chiral tetrominoes (L,S) in 2-space;
6 pairs of chiral pentominoes in 3-space.
		

Links

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

Crossrefs

Cf. A045648, A045649, A036364.

Programs

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} ]

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).