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 A196570 LAN party numbers. 1
 0, 4, 6, 18, 48, 118, 314, 806, 2082, 5402, 13946, 36102, 93378, 241518, 624810, 1616142, 4180594, 10814158, 27973298, 72359966, 187176434, 484177358, 1252442706, 3239746862, 8380393330, 21677923822, 56075218194, 145052181998, 375212720786, 970578896942 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) gives the number of solutions to the following problem: n couples, each consisting of 1 boy and 1 girl, are at a LAN party with a 2 x n table, where pairs of index cards labeled "B" and "G" have been laid out on either adjacent or opposite seats. Each boy takes a seat at a chair marked "B" and his date takes the corresponding "G" chair either next to him or across from him. Additionally, each boy finds that he is either across from or next to at least one other boy. How many ways are there for the host to arrange the pairs of cards that result in such an assignment? LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 Jonathan Daniel Kilgallin, LAN party numbers Index entries for linear recurrences with constant coefficients, signature (1,4,2,-4,-2,2). FORMULA m(1) = 0; m(2) = 2; m(3) = 4; m(4) = 10. x(1) = 2; x(2) = 0; x(3) = 6; x(4) = 12. a(1) = 0; a(2) = 4; a(3) = 6; a(4) = 18. m(n) = 2*a(n-3) + m(n-1) + x(n-1). x(n) = 2*a(n-3) + 2*a(n-4) + m(n-1) + m(n-2) + x(n-2) + 2*x(n-3). a(n) = 2*a(i-2) + x(n-2) + m(n). G.f.: -2*x^2*(2+x-2*x^2-x^3+x^4)/ ((1+x)*(2*x^5-4*x^4+2*x^2+2*x-1)). - Alexander R. Povolotsky, Oct 05 2011 EXAMPLE for n = 3 the a(3)=6 solutions are G-B G | G-B B +++++ G-B B | G-B G +++++ G G G | | | B B B +++++ G B-G | B B-G +++++ B B-G | G B-G +++++ B B B | | | G G G MATHEMATICA CoefficientList[Series[-2*x^2*(2 + x - 2*x^2 - x^3 + x^4)/((1 + x)*(2*x^5 - 4*x^4 + 2*x^2 + 2*x - 1)), {x, 0, 50}], x] (* G. C. Greubel, Feb 22 2017 *) PROG (C#) public static long a(int n) { long[] M = new long[n+1]; long[] X = new long[n+1]; long[] S = new long[n+1]; M = 0; M = 2; M = 4; M = 10; X = 2; X = 0; X = 6; X = 12; S = 0; S = 4; S = 6; S = 18; for (int i = 5; i <= n; i++) { M[i] = 2 * S[i-3] + M[i-1] + X[i-1]; X[i] = 2 * S[i - 3] + 2 * S[i - 4] + M[i - 1] + M[i - 2] + X[i - 2] + 2 * X[i - 3]; S[i] = 2 * S[i - 2] + X[i - 2] + M[i]; } return S[n]; } (PARI) x='x+O('x^50); Vec(-2*x^2*(2 + x - 2*x^2 - x^3 + x^4)/((1 + x)*(2*x^5 - 4*x^4 + 2*x^2 + 2*x - 1)) \\ G. C. Greubel, Feb 22 2017 CROSSREFS Sequence in context: A218065 A005959 A057393 * A274992 A317584 A012928 Adjacent sequences: A196567 A196568 A196569 * A196571 A196572 A196573 KEYWORD nonn,easy AUTHOR Jonathan Daniel Kilgallin, Oct 04 2011 STATUS approved

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Last modified March 29 12:09 EDT 2023. Contains 361599 sequences. (Running on oeis4.)