A371978 - OEIS

%I #20 Apr 16 2024 08:27:32

%S 1,3,8,22,61,174,504,1478,4374,13035,39062,117585,355279,1076845,

%T 3272692,9969385,30430982,93055869,285013326,874193006,2684778104,

%U 8254967674,25408703236,78283452265,241403160254,745024894092,2301051484006,7111897305089,21995136183906

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

%H Alois P. Heinz, <a href="/A371978/b371978.txt">Table of n, a(n) for n = 0..2011</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wazir_(chess)">Wazir (chess)</a>

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

%F From _Vaclav Kotesovec_, Apr 16 2024: (Start)

%F 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).

%F 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)

%e a(2) = 8:

%e   +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+

%e   | W . | | W . | | W . | | . W | | . W | | . W | | . . | | . . |

%e   | . W | | . . | | . . | | W . | | . . | | . . | | W . | | . W |

%e   | . . | | W . | | . W | | . . | | W . | | . W | | . W | | W . |

%e   +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ .

%p b:= proc(n, l) option remember; `if`(n=0, 1,

%p       add(`if`(Bits[And](j, l)>0, 0, expand(b(n-1, j)*

%p       x^add(i, i=Bits[Split](j)))), j=[0, 1, 2, 4, 5]))

%p     end:

%p a:= n-> coeff(b(n, 0), x, n):

%p seq(a(n), n=0..30);

%Y Main diagonal of A371967.

%Y Cf. A201511, A371979.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Apr 14 2024