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 A222715 The number of binary pattern classes in the (2,n)-rectangular grid with 5 '1's and (2n-5) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other. 4
 2, 14, 66, 198, 508, 1092, 2156, 3876, 6606, 10626, 16478, 24570, 35672, 50344, 69624, 94248, 125562, 164502, 212762, 271502, 342804, 428076, 529828, 649740, 790790, 954954, 1145718, 1365378, 1617968, 1906128, 2234480, 2606032, 3026034, 3497886, 4027506 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 LINKS Vincenzo Librandi, Table of n, a(n) for n = 3..1000 Index entries for linear recurrences with constant coefficients, signature (3,0,-8,6,6,-8,0,3,-1). FORMULA a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6) -4*(2*n^2-22*n+63)*(-1)^n, with n>8, a(3)=2, a(4)=14, a(5)=66, a(6)=198, a(7)=508, a(8)=1092. From Bruno Berselli, May 29 2013: (Start) G.f.: 2*x^3*(1+4*x+12*x^2+8*x^3+7*x^4)/((1+x)^3*(1-x)^6). a(n) = 3*a(n-1) -8*a(n-3) +6*a(n-4) +6*a(n-5) -8*a(n-6) +3*a(n-8) -a(n-9), with n>11. a(n) = (n-2)*(n-1)*(8*n^3-16*n^2+6*n-15*(-1)^n+15)/120. (End) a(n) = (1/4)*(binomial(2*n,5) + 2*binomial(n-1,2)*(1/2)*(1-(-1)^n)). [Yosu Yurramendi and María Merino, Aug 21 2013] MATHEMATICA Table[(n - 2) (n - 1) ((8 n^3 - 16 n^2 + 6 n - 15 (-1)^n + 15)/120), {n, 3, 40}] (* Bruno Berselli, May 30 2013 *) LinearRecurrence[{3, 0, -8, 6, 6, -8, 0, 3, -1}, {2, 14, 66, 198, 508, 1092, 2156, 3876, 6606}, 50] (* T. D. Noe, Jun 14 2013 *) CoefficientList[Series[2 (1 + 4 x + 12 x^2 + 8 x^3 + 7 x^4) / ((1 + x)^3 (1 - x)^6), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 04 2013 *) PROG (Magma) m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x^3*(1+4*x+12*x^2+8*x^3+7*x^4)/((1+x)^3*(1-x)^6))); (R) a <- vector() for(n in 1:40) a[n] <- (1/4)*(choose(2*(n+2), 5) + 2*choose(n+1, 2)*(1/2)*(1-(-1)^n)) a [Yosu Yurramendi and María Merino, Aug 21 2013] (Magma) [(1/4)*(Binomial(2*n, 5) + 2*Binomial(n-1, 2)*(1/2)*(1-(-1)^n)): n in [3..40]]; // Vincenzo Librandi, Sep 04 2013 CROSSREFS Cf. A226048. Sequence in context: A266590 A196977 A254197 * A197162 A109869 A197777 Adjacent sequences: A222712 A222713 A222714 * A222716 A222717 A222718 KEYWORD nonn,easy AUTHOR Yosu Yurramendi, May 29 2013 STATUS approved

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Last modified September 24 17:02 EDT 2023. Contains 365579 sequences. (Running on oeis4.)