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%I #49 Jun 06 2019 20:20:53
%S 1,4,10,40,136,544,2080,8320,32896,131584,524800,2099200,8390656,
%T 33562624,134225920,536903680,2147516416,8590065664,34359869440,
%U 137439477760,549756338176,2199025352704,8796095119360,35184380477440,140737496743936,562949986975744
%N Number of reversible strings with n beads of 4 colors.
%C 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
%H Colin Barker, <a href="/A032121/b032121.txt">Table of n, a(n) for n = 0..1000</a>
%H Christian Barrientos, Sarah Minion, <a href="https://digitalcommons.georgiasouthern.edu/tag/vol6/iss1/4">Series-Parallel Operations with Alpha-Graphs</a>, Theory and Applications of Graphs (2019) Vol. 6, Issue 1, Article 4.
%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,-16).
%F "BIK" (reversible, indistinct, unlabeled) transform of 4, 0, 0, 0, ...
%F a(n) = (4^m+3*2^m+(-2)^m)/8, where m = n+1. - _Frank Ruskey_, Jul 14 2002
%F G.f.: (1-10x^2) / ((1-4x)*(1-4x^2)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009; corrected by _R. J. Mathar_, Sep 16 2009 [Adapted to offset 0 by _Robert A. Russell_, Nov 10 2018]
%F From _Colin Barker_, Nov 25 2017: (Start)
%F a(n) = 2^(n-2) * (3 + (-1)^(1+n) + 2^(1+n)).
%F a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3) for n>2.
%F (End)
%F a(n) = (4^n + 4^floor((n+1)/2)) / 2 = (A000302(n) + A056450(n)) / 2. - _Robert A. Russell_ and _Danny Rorabaugh_, Jun 22 2018
%F E.g.f.: (1/4)*exp(-2*x)*(- 1 + 3*exp(4*x) + 2*exp(6*x)). - _Stefano Spezia_, Nov 12 2018
%e a(2) = 10 = |{000, 110,101,011, 220,202,022, 330,303,033}|.
%t k = 4; Table[(k^n + k^Ceiling[n/2])/2, {n, 0, 30}] (* _Robert A. Russell_, Nov 25 2017 *)
%t LinearRecurrence[{4, 4, -16}, {1, 4, 10}, 31] (* _Robert A. Russell_, Nov 10 2018 *)
%t CoefficientList[Series[1/4 E^(-2 x) (-1 + 3 E^(4 x) + 2 E^(6 x)), {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* _Stefano Spezia_, Nov 12 2018 *)
%o (PARI) Vec((1-10*x^2) / ((1 - 2*x)*(1 + 2*x)*(1 - 4*x)) + O(x^40)) \\ _Colin Barker_, Nov 25 2017
%Y Column 4 of A277504.
%Y Cf. A000302 (oriented), A032087(n>1) (chiral), A056450 (achiral).
%K nonn,easy
%O 0,2
%A _Christian G. Bower_
%E a(0) = 1 prepended by _Robert A. Russell_, Nov 10 2018
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