A218073 Number of profiles in domino tiling of a 2*n checkboard.

0, 1, 2, 9, 12, 50, 60, 245, 280, 1134, 1260, 5082, 5544, 22308, 24024, 96525, 102960, 413270, 437580, 1755182, 1847560, 7407036, 7759752, 31097794, 32449872, 130007500, 135207800, 541574100, 561632400, 2249204040, 2326762800, 9316746045, 9617286240, 38504502630

Offset: 0

Author

Michel Marcus, Oct 20 2012

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• T. C. Wu, Counting the Profiles in Domino Tiling, The Fibonacci Quarterly, Volume 21, Number 4, November 1983, pp. 302-304.

Crossrefs

Cf. A005430 (bisection).

Programs

Maple

a:= proc(n) option remember;
      `if`(n<3, n, (n*(5-7*n)*a(n-1) +4*(n-2)*(7*n+16)*a(n-3)
      +(24-12*n+172*n^2)*a(n-2))/ ((n+1)*(43*n-89)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 20 2012

Mathematica

a[n_] := n/2*Binomial[n + Mod[n, 2], (n + Mod[n, 2])/2]; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Feb 22 2013, after Joerg Arndt *)

Maxima

a[0]:0$a[1]:1$a[2]:2$
a[n]:=(n*(5-7*n)*a[n-1] +4*(n-2)*(7*n+16)*a[n-3]+(24-12*n+172*n^2)*a[n-2])/ ((n+1)*(43*n-89))$
makelist(a[n] ,n,0,40); /* Martin Ettl, Oct 21 2012 */

PARI

a(n) = n/2 * binomial(n+(n%2),(n+n%2)/2); /* Joerg Arndt, Oct 21 2012 */

Formula

If n is even, a(n) = binomial(n, n/2)*n/2.

If n is odd, a(n) = binomial(n + 1, (n + 1)/2)*n/2.