A182562 Number of ways to place k non-attacking semi-knights on an n x n chessboard, sum over all k>=0

2, 16, 288, 11664, 1458000, 506250000, 414720000000, 869730877440000, 5045702916833280000, 77297454895962562560000, 3017525202366485003182080000, 307389127582207654481154908160000, 83016370640108703579427655610531840000, 58770343311359208383258439665073059266560000

Offset

1,1

Comments

Semi-knight is a semi-leaper [1,2]. Moves of a semi-knight are allowed only in [2,1] and [-2,-1]. See also semi-bishops (A187235).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..60

V. Kotesovec, Non-attacking chess pieces

Formula

a(n) = F(n/2+2)^(n+2)*prod(j=1,n/2-1,F(j+2)^4) if n is even, F((n+1)/2+2)^((n+1)/2)*F((n-1)/2+2)^((n-1)/2)*prod(j=1,(n-1)/2,F(j+2)^4) if n is odd, where F(n) = A000045(n) is the n-th Fibonacci number.

a(n) is asymptotic to C^4*((1+sqrt(5))/2)^((n+2)*(n+4))/5^(3/2*(n+2)), where C=1.226742010720353244... is Fibonacci Factorial Constant, see A062073.

Mathematica

Table[If[EvenQ[n], Fibonacci[n/2+2]^(n+2)*Product[Fibonacci[j+2]^4, {j, 1, n/2-1}], Fibonacci[(n+1)/2+2]^((n+1)/2)*Fibonacci[(n-1)/2+2]^((n-1)/2)*Product[Fibonacci[j+2]^4, {j, 1, (n-1)/2}]], {n, 1, 20}]

Crossrefs

Cf. A067962, A067966, A063443, A006506, A067965, A066864, A067963, A067964, A182563

Sequence in context: A009549 A254744 A009795 * A112722 A223664 A189611

Adjacent sequences: A182559 A182560 A182561 * A182563 A182564 A182565

Keyword

nonn

Author

Vaclav Kotesovec, May 05 2012

Status

approved