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 A032091 Number of reversible strings with n-1 beads of 2 colors. 4 beads are black. String is not palindromic. 10
 2, 6, 16, 32, 60, 100, 160, 240, 350, 490, 672, 896, 1176, 1512, 1920, 2400, 2970, 3630, 4400, 5280, 6292, 7436, 8736, 10192, 11830, 13650, 15680, 17920, 20400, 23120, 26112, 29376, 32946, 36822, 41040, 45600, 50540, 55860, 61600, 67760, 74382, 81466, 89056 (list; graph; refs; listen; history; text; internal format)
 OFFSET 6,1 COMMENTS Also, number of 4-element subsets of the set {1,...,n-1} whose elements sum up to an odd integer, i.e., 4th column of the triangle A159916, cf. there. - M. F. Hasler, May 01 2009 Also, if the offset is changed to 3, so that a(3)=2, a(n) = number of non-equivalent (mod D_3) ways to place 2 indistinguishable points on a triangular grid of side n so that they are not adjacent. - Heinrich Ludwig, Mar 23 2014 Also, the number of binary strings of length n with exactly one pair of consecutive 0s and exactly three pairs of consecutive 1s. - Jeremy Dover, Jul 07 2016 LINKS Colin Barker, Table of n, a(n) for n = 6..1000 C. G. Bower, Transforms (2) Elizabeth Wilmer, Notes on Stephan's conjectures 72, 73 and 74 Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1) FORMULA "BHK[ 5 ]" (reversible, identity, unlabeled, 5 parts) transform of 1, 1, 1, 1... From M. F. Hasler, May 01 2009: (Start) G.f.: -2*x^6 / ((x-1)^5*(x+1)^2). - Corrected by Colin Barker, Mar 07 2015 a(n) = [(n-5)(n-3)(n-1)^2 + (6n-15) X[2Z](n)]/48, where X[2Z] is the characteristic function of 2Z. (End) From Colin Barker, Mar 07 2015: (Start) a(n) = (n^4-10*n^3+32*n^2-32*n)/48 if n is even. a(n) = (n^4-10*n^3+32*n^2-38*n+15)/48 if n is odd. (End) a(n) = (2*n^4 - 20*n^3 + 64*n^2 + 6*(-1)^n*n - 70*n - 15*(-1)^n + 15)/96. - Ilya Gutkovskiy, Jul 08 2016 MATHEMATICA Table[If[EvenQ[n], (n^4-10n^3+32n^2-32n)/48, (n^4-10n^3+32n^2-38n+15)/48], {n, 6, 50}] (* or *) LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {2, 6, 16, 32, 60, 100, 160}, 50] (* Harvey P. Dale, Apr 11 2016 *) PROG (PARI) A032091(n)=polcoeff(2/(1-x)^5/(1+x)^2+O(x^(n-5)), n-6) A032091(n)=((n-5)*(n-3)*(n-1)^2+if(n%2==0, 6*n-15))/48 \\ M. F. Hasler, May 01 2009 CROSSREFS a(n+6) = 2*A002624(n), A239572. Fourth column of A274228. - Jeremy Dover, Jul 07 2016 Sequence in context: A192735 A218212 A171218 * A182994 A235792 A192706 Adjacent sequences:  A032088 A032089 A032090 * A032092 A032093 A032094 KEYWORD nonn,easy AUTHOR STATUS approved

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