This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357723 #52 Jun 02 2025 16:49:53 %S A357723 0,0,0,5,21,63,135,270,462,770,1170,1755,2475,3465,4641,6188,7980, %T A357723 10260,12852,16065,19665,24035,28875,34650,40986,48438,56550,65975, %U A357723 76167,87885,100485,114840,130200,147560,166056,186813,208845,233415,259407,288230,318630 %N A357723 Number of ways to place a non-attacking black king and white king on an n X n board, up to rotation and reflection. %C A357723 Rotations and reflections of placements are not counted. (If they were then see A035286.) %C A357723 a(8)=462 is the number of states in the KvK endgame in an eightfold-reducing chess tablebase on 8 X 8 boards. %C A357723 When kings are unlabeled, see A279111. The ratio a(n)/A279111(n) is bounded in the interval [1, 2] and converges to 2, because the number of placements in which the kings' positions can be swapped by an automorphism is O(n^2), while the sequence itself is O(n^4). %C A357723 When there are pawns on the board and the position is only equivalent under reflection in the x axis, see A357740. %C A357723 A quasipolynomial of degree 4 and period 2. - _Charles R Greathouse IV_, Feb 02 2023 %H A357723 <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 A357723 a(n) = n^4/8 - (5/8)*n^2 + 1/2 if n is odd, else n^4/8 - (7/8)*n^2 + (3/4)*n. %F A357723 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 A357723 a(n) = n^4/8 - (3/4)*n^2 + (3/8)*n + 1/4 + (-(1/8)*n^2 + (3/8)*n - 1/4)*(-1)^n. %F A357723 a(n) = (n^4 + (2*(n mod 2)-7)*n^2 + 6*(1-(n mod 2))*n + (n mod 2)*4)/8. %F A357723 a(n) = (n-2)*(n-1)*(n^2 + 3*n + 2*(n mod 2))/8. %F A357723 G.f.: x^3*(3*x^3 - 11*x^2 - 11*x - 5)/((x+1)^3*(x-1)^5). %F A357723 E.g.f.: (x*(x^3 + 6*x^2 - 4)*cosh(x) + (x^4 + 6*x^3 + 2*x^2 + 4)*sinh(x))/8. - _Stefano Spezia_, Jan 28 2023 %e A357723 For n=3, the a(3) = 5 solutions are %e A357723 ... ... ..b b.. .b. %e A357723 ... ..b ... ... ... %e A357723 w.b w.. w.. .w. .w. %o A357723 (Python) %o A357723 a=(lambda n: ((n-2)*(n-1)*(n**2+3*n+n%2*2)//8)) %o A357723 (PARI) a(n)=(n-2)*(n-1)*(n^2+3*n+n%2*2)\8 \\ _Charles R Greathouse IV_, Feb 02 2023 %Y A357723 Cf. A035286, A279111, A357740. %K A357723 nonn,easy %O A357723 0,4 %A A357723 _Natalia L. Skirrow_, Oct 10 2022