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A032121 Number of reversible strings with n beads of 4 colors. 8
4, 10, 40, 136, 544, 2080, 8320, 32896, 131584, 524800, 2099200, 8390656, 33562624, 134225920, 536903680, 2147516416, 8590065664, 34359869440, 137439477760, 549756338176, 2199025352704, 8796095119360, 35184380477440, 140737496743936, 562949986975744 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Also the number of 4-ary strings of length m = n+1 with number of 1's, 2's and 3's all even. Bijective proof anyone? - Frank Ruskey, Jul 14 2002

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

C. G. Bower, Transforms (2)

Index entries for linear recurrences with constant coefficients, signature (4,4,-16).

FORMULA

"BIK" (reversible, indistinct, unlabeled) transform of 4, 0, 0, 0...

a(n) = (4^m+3*2^m+(-2)^m)/8, where m = n+1. - Frank Ruskey, Jul 14 2002

G.f.: 2*x*(2-3*x-8*x^2)/((2*x+1)*(2*x-1)*(4*x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009; corrected by R. J. Mathar, Sep 16 2009

From Colin Barker, Nov 25 2017: (Start)

a(n) = 2^(n-2) * (3 + (-1)^(1+n) + 2^(1+n)) for n>0.

a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3) for n>3.

(End)

EXAMPLE

a(2) = 10 = |{000, 110,101,011, 220,202,022, 330,303,033}|.

MATHEMATICA

k=4; Table[(k^n+k^Ceiling[n/2])/2, {n, 1, 30}] (* Robert A. Russell, Nov 25 2017 *)

PROG

(PARI) Vec(2*x*(2 - 3*x - 8*x^2) / ((1 - 2*x)*(1 + 2*x)*(1 - 4*x)) + O(x^40)) \\ Colin Barker, Nov 25 2017

CROSSREFS

Column 4 of A277504.

Sequence in context: A149208 A149209 A053792 * A149210 A149211 A149212

Adjacent sequences:  A032118 A032119 A032120 * A032122 A032123 A032124

KEYWORD

nonn,easy

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified February 20 01:25 EST 2018. Contains 299357 sequences. (Running on oeis4.)