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A357740 Number of non-equivalent ways under symmetry in one axis that 2 non-attacking kings of different colors can be placed on an n X n board.

%I A357740 #30 Jun 02 2025 16:49:53
%S A357740 0,0,17,78,234,520,1035,1806,2996,4608,6885,9790,13662,18408,24479,
%T A357740 31710,40680,51136,63801,78318,95570,115080,137907,163438,192924,
%U A357740 225600,262925,303966,350406,401128,458055,519870,588752,663168,745569,834190,931770,1036296,1150811,1273038
%N A357740 Number of non-equivalent ways under symmetry in one axis that 2 non-attacking kings of different colors can be placed on an n X n board.
%C A357740 The number of king positions over which you iterate when making tablebases of positions containing pawns, wherein it is only equivalent under reflection in the x axis.
%H A357740 Andrew Howroyd, <a href="/A357740/b357740.txt">Table of n, a(n) for n = 1..1000</a>
%H A357740 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F A357740 a(n) = n^4/2 - 4*n^2 + (9/2)*n - 1 if n is odd else n^4/2 - (9/2)*n^2 + 6*n - 2;
%F A357740 a(n) = n^4/2 - (17/4)*n^2 + (21/4)*n - 3/2 + (-1)^n*(-(1/4)*n^2 + (3/4)*n - 1/2);
%F A357740 a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8);
%F A357740 a(n) = (n-2)*(n-1)*((n+3)*n - 2 + (n mod 2))/2.
%F A357740 G.f.: x^3*(17 + 44*x + 44*x^2 - 2*x^3 - 5*x^4 - 2*x^5)/((1 - x)^5*(1 + x)^3). - _Andrew Howroyd_, Dec 31 2022
%F A357740 E.g.f.: 2+(e^x*(2*x^4 + 12*x^3 - 3*x^2 + 6*x - 6) - e^(-x)*(x^2 + 2*x + 2))/4 = (cosh(x)*(x^4 + 6*x^3 - 2*x^2 + 2*x - 4) + sinh(x)*(x^4 + 6*x^3 - x^2 + 4*x - 2))/2 + 2.
%e A357740 For n=3, the a(3)=17 solutions are
%e A357740    |   |  K|   |   |  K|k  |k  |k K| K |K  |K  | K |k  |k  | k | k |
%e A357740    |  K|   |k  |k K|k  |   |  K|   |   |   |   |   |   |   |   |   |
%e A357740 k K|k  |k  |  K|   |   |  K|   |   |k  |k  | k | k | K |K  |K  | K |
%o A357740 (Python) a=(lambda n: (n-2)*(n-1)*((n+3)*n-2+n%2)//2)
%o A357740 (PARI) a(n) = {(n-2)*(n-1)*((n+3)*n - 2 + (n % 2))/2} \\ _Andrew Howroyd_, Dec 31 2022
%Y A357740 Cf. A035286 (no symmetry), A357723 (8-fold symmetry).
%K A357740 nonn,easy
%O A357740 1,3
%A A357740 _Natalia L. Skirrow_, Oct 11 2022