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%I #24 Mar 26 2023 18:38:07
%S 80,1847,26040,281571,2580754,21137959,159636030,1134127305,
%T 7683664202,50123713793,317076250136,1955475353217,11806000507544,
%U 70004699407151,408747986045656,2355077855615435,13413115039118042,75623103424916527
%N Number of ways to place 4n nonattacking kings on an 8 X 2n chessboard.
%H Vincenzo Librandi, <a href="/A173782/b173782.txt">Table of n, a(n) for n = 1..1000</a>
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F G.f.: x*(22500*x^16 -382125*x^15 +2723005*x^14 -10917322*x^13 +27938661*x^12 -48873227*x^11 +60780149*x^10 -54895129*x^9 +36368733*x^8 -17776175*x^7 +6499001*x^6 -1854479*x^5+446565*x^4 -94300*x^3 +15732*x^2 -1673*x+80) / ((1-x) *(x^2-4*x+1) *(x^3-6*x^2+5*x-1) *(4*x-1) *(5*x-1)^2 *(3*x^2-5*x+1)^2 *(5*x^2-5*x+1)^2).
%F Recurrence: a(n) = 44a(n-1) -887a(n-2) +10855a(n-3) -90083a(n-4) +536398a(n-5) -2365292a(n-6) +7860674a(n-7) -19852652a(n-8) +38152568a(n-9) -55523880a(n-10) +60518766a(n-11) -48502595a(n-12) +27783210a(n-13) -10888525a(n-14) +2721025a(n-15) -382125a(n-16) +22500a(n-17), n>17.
%F a(n) = (-12505804889/302760 +7963567/2610*n)*5^n +3872/3*4^n -1/24 +(135343*sqrt(3)/18 -234421/18)*(2 -sqrt(3))^n -(135343*sqrt(3)/18 +234421/18)*(2 +sqrt(3))^n +(33301/5 -74461*sqrt(5)/25 +(141*sqrt(5)/25 -63/5)*n)*((5 -sqrt(5))/2)^n +(74461*sqrt(5)/25 +33301/5 - (141*sqrt(5)/25 + 63/5)*n)*((5 +sqrt(5))/2)^n + (4306740/169 - 1194474*sqrt(13)/169 + (139103/117 - 501541*sqrt(13)/1521)*n)*((5 -sqrt(13))/2)^n +(1194474*sqrt(13)/169 +4306740/169 +(501541*sqrt(13)/1521 +139103/117)*n)*((5 +sqrt(13))/2)^n +72*(b*(3504697*c - 11380560) -11380560*c +36953816)/(142129*(a - b)*(a - c))*a^n +72*(a*(3504697*c - 11380560) - 8*(1422570*c - 4619227))/(142129*(a - b)*(c - b))*b^n +72*(a*(3504697*b - 11380560) -8*(1422570*b - 4619227))/(142129*(a - c)*(b - c))*c^n, where: a=2-2*sin(Pi/14), b=2+2*sin(3*Pi/14), c=2-2*cos(Pi/7). - _Vaclav Kotesovec_, added Mar 01 2010, updated Mar 29 2010.
%t CoefficientList[Series[(22500 x^16 - 382125 x^15 + 2723005 x^14 - 10917322 x^13 + 27938661 x^12 - 48873227 x^11 + 60780149 x^10 - 54895129 x^9 + 36368733 x^8 - 17776175 x^7 + 6499001 x^6 - 1854479 x^5 + 446565 x^4 - 94300 x^3 + 15732 x^2 - 1673 x + 80) / ((1 - x) (x^2 - 4 x + 1) (x^3 - 6 x^2 + 5 x - 1) (4 x - 1) (5 x - 1)^2 (3 x^2 - 5 x + 1)^2 (5 x^2 - 5 x + 1)^2),{x, 0, 50}], x] (* _Vincenzo Librandi_, May 30 2013 *)
%Y Column k=4 of A350819.
%Y Cf. A061594, A061593, A018807.
%K nonn,easy
%O 1,1
%A _Vaclav Kotesovec_, Feb 24 2010
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