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A032091 Number of reversible strings with n-1 beads of 2 colors. 4 beads are black. String is not palindromic. 7
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

Christian G. Bower

STATUS

approved

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Last modified December 10 03:42 EST 2016. Contains 278993 sequences.