A045649 Number of chiral n-ominoes in n-1 space.

1, 0, 0, 1, 1, 1, 2, 5, 9, 15, 31, 70, 146, 300, 656, 1471, 3258, 7245, 16400, 37461, 85773, 197365, 457297, 1065070, 2489750, 5842741, 13766775, 32552087, 77208432, 183670145, 438176307, 1048092760, 2513081101

Offset: 1

Author

Robert A. Russell

Keywords

• easy
• nice
• nonn

Comments

Lunnon's DR(n,n-1)-DE(n,n-1). Knuth describes methodology for a similar enumeration, that of free trees with n nodes.

References

• D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, pp. 386-88.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
• W. F. Lunnon, Counting Multidimensional Polyominoes, Computer Journal, Vol. 18 (1975), pp. 366-67.

Crossrefs

Cf. A045648, A000081, A004111.

Programs

Mathematica

s[ n_, k_ ] := s[ n, k ]=c[ n+1-k ]+If[ n<2k, 0, s[ n-k, k ](-1)^k ]; c[ 1 ]=1; c[ n_ ] := c[ n ]=Sum[ c[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ c[ i ]-Sum[ c[ j ]c[ i-j ], {j, 1, i/2} ]+If[ OddQ[ i ], 0, c[ i/2 ](c[ i/2 ]+(-1)^(i/2))/2 ], {i, 1, 33} ]

Formula

G.f.: C(x)-C^2(x)/2+C(-x^2)/2 where C(x) is g.f. for same sequence with one cell labeled, A045648.

a(n) ~ c * d^n / n^(5/2), where d = 2.58968405406171542574769690513208346256... and c = 0.36257350770010314582973624284... . - Vaclav Kotesovec, Feb 29 2016