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%I #38 Oct 12 2022 08:41:53
%S 1,5,22,98,436,1940,8632,38408,170896,760400,3383392,15054368,
%T 66984256,298045760,1326151552,5900697728,26255094016,116821771520,
%U 519797274112,2312832639488,10290925106176,45789365703680,203739313027072,906535983515648,4033622560116736
%N Number of words of length n over the alphabet {1,2,3,4,5} such that 1 is not followed by an odd letter.
%C Appears to be Kekulé numbers for certain benzenoids (see the Cyvin-Gutman book for details).
%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78).
%H Andrew Howroyd, <a href="/A123347/b123347.txt">Table of n, a(n) for n = 0..500</a>
%H D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3, example 17.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4, 2).
%F From _Klaus Brockhaus_, Oct 03 2009: (Start)
%F Inverse binomial transform of A138395.
%F a(n) = ((2+sqrt(6))^(n+1) + (2-sqrt(6))^(n+1))/4.
%F a(n) = 4*a(n-1) + 2*a(n-2) for n > 1.
%F G.f.: (1 + x)/(1 - 4*x - 2*x^2).
%F (End)
%F a(n) = A090017(n+1)+A090017(n). - _R. J. Mathar_, Aug 04 2019
%e a(2) = 22 because all 25 words of length 2 are included except 11, 13 and 15.
%p seq(coeff(series((1+x)/(1-4*x-2*x^2),x,n+1), x, n), n = 0 .. 25); # _Muniru A Asiru_, Nov 27 2018
%t LinearRecurrence[{4, 2}, {1, 5}, 30] (* _Amiram Eldar_, Nov 26 2018 *)
%o (PARI) Vec((1 + x)/(1 - 4*x - 2*x^2) + O(x^30)) \\ _Andrew Howroyd_, Nov 25 2018
%o (Magma) I:=[1, 5]; [n le 2 select I[n] else 4*Self(n-1) + 2*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Nov 29 2018
%o (Sage) s=((1+x)/(1-4*x-2*x^2)).series(x, 50); s.coefficients(x, sparse=False) # _G. C. Greubel_, Nov 29 2018
%Y Cf. A138395.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Oct 10 2006
%E Edited and new name by _Armend Shabani_ and _Andrew Howroyd_, Nov 25 2018
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