A371978 Number of ways of placing n non-attacking wazirs on a 3 X n board.

1, 3, 8, 22, 61, 174, 504, 1478, 4374, 13035, 39062, 117585, 355279, 1076845, 3272692, 9969385, 30430982, 93055869, 285013326, 874193006, 2684778104, 8254967674, 25408703236, 78283452265, 241403160254, 745024894092, 2301051484006, 7111897305089, 21995136183906

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2011

Wikipedia, Wazir (chess)

Formula

a(n) = A371967(n,n).

From Vaclav Kotesovec, Apr 16 2024: (Start)

Recurrence: (n+1)*(72*n^4 - 700*n^3 + 2288*n^2 - 2803*n + 796)*a(n) = 2*(144*n^5 - 1328*n^4 + 3814*n^3 - 3083*n^2 - 1479*n + 1194)*a(n-1) - 2*(72*n^5 - 700*n^4 + 2050*n^3 - 1979*n^2 + 409*n + 16)*a(n-2) - 4*(36*n^5 - 368*n^4 + 1437*n^3 - 2421*n^2 + 1398*n + 95)*a(n-3) - (72*n^5 - 772*n^4 + 2404*n^3 - 1365*n^2 - 4749*n + 5704)*a(n-4) + 2*(72*n^5 - 808*n^4 + 2858*n^3 - 3067*n^2 - 1494*n + 2666)*a(n-5) - (n-6)*(72*n^4 - 412*n^3 + 620*n^2 - 39*n - 347)*a(n-6).

a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = (188 + 12*sqrt(93))^(1/3)/6 + 14/(3*(188 + 12*sqrt(93))^(1/3)) + 4/3 and c = 11/6 + (1465336244224 - 5597165568*sqrt(93))^(1/3)/5952 + ((23080523 + 88161*sqrt(93))/2)^(1/3) / (12*31^(2/3)). (End)

Example

a(2) = 8:
  +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+
  | W . | | W . | | W . | | . W | | . W | | . W | | . . | | . . |
  | . W | | . . | | . . | | W . | | . . | | . . | | W . | | . W |
  | . . | | W . | | . W | | . . | | W . | | . W | | . W | | W . |
  +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ .

Maple

b:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(Bits[And](j, l)>0, 0, expand(b(n-1, j)*
      x^add(i, i=Bits[Split](j)))), j=[0, 1, 2, 4, 5]))
    end:
a:= n-> coeff(b(n, 0), x, n):
seq(a(n), n=0..30);

Crossrefs

Main diagonal of A371967.

Cf. A201511, A371979.

Sequence in context: A121449 A025566 A027036 * A018040 A018041 A073357

Adjacent sequences: A371975 A371976 A371977 * A371979 A371980 A371981

Keyword

nonn

Author

Alois P. Heinz, Apr 14 2024

Status

approved