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

%I A371978 #20 Apr 16 2024 08:27:32
%S A371978 1,3,8,22,61,174,504,1478,4374,13035,39062,117585,355279,1076845,
%T A371978 3272692,9969385,30430982,93055869,285013326,874193006,2684778104,
%U A371978 8254967674,25408703236,78283452265,241403160254,745024894092,2301051484006,7111897305089,21995136183906
%N A371978 Number of ways of placing n non-attacking wazirs on a 3 X n board.
%H A371978 Alois P. Heinz, <a href="/A371978/b371978.txt">Table of n, a(n) for n = 0..2011</a>
%H A371978 Wikipedia, <a href="https://en.wikipedia.org/wiki/Wazir_(chess)">Wazir (chess)</a>
%F A371978 a(n) = A371967(n,n).
%F A371978 From _Vaclav Kotesovec_, Apr 16 2024: (Start)
%F A371978 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 A371978 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 A371978 a(2) = 8:
%e A371978   +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+
%e A371978   | W . | | W . | | W . | | . W | | . W | | . W | | . . | | . . |
%e A371978   | . W | | . . | | . . | | W . | | . . | | . . | | W . | | . W |
%e A371978   | . . | | W . | | . W | | . . | | W . | | . W | | . W | | W . |
%e A371978   +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ +-----+ .
%p A371978 b:= proc(n, l) option remember; `if`(n=0, 1,
%p A371978       add(`if`(Bits[And](j, l)>0, 0, expand(b(n-1, j)*
%p A371978       x^add(i, i=Bits[Split](j)))), j=[0, 1, 2, 4, 5]))
%p A371978     end:
%p A371978 a:= n-> coeff(b(n, 0), x, n):
%p A371978 seq(a(n), n=0..30);
%Y A371978 Main diagonal of A371967.
%Y A371978 Cf. A201511, A371979.
%K A371978 nonn
%O A371978 0,2
%A A371978 _Alois P. Heinz_, Apr 14 2024