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A123347
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Number of words of length n over the alphabet {1,2,3,4,5} such that 1 is not followed by an odd letter.
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9
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1, 5, 22, 98, 436, 1940, 8632, 38408, 170896, 760400, 3383392, 15054368, 66984256, 298045760, 1326151552, 5900697728, 26255094016, 116821771520, 519797274112, 2312832639488, 10290925106176, 45789365703680, 203739313027072, 906535983515648, 4033622560116736
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OFFSET
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0,2
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COMMENTS
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Appears to be Kekulé numbers for certain benzenoids (see the Cyvin-Gutman book for details).
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REFERENCES
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S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78).
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LINKS
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FORMULA
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Inverse binomial transform of A138395.
a(n) = ((2+sqrt(6))^(n+1) + (2-sqrt(6))^(n+1))/4.
a(n) = 4*a(n-1) + 2*a(n-2) for n > 1.
G.f.: (1 + x)/(1 - 4*x - 2*x^2).
(End)
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EXAMPLE
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a(2) = 22 because all 25 words of length 2 are included except 11, 13 and 15.
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MAPLE
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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
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MATHEMATICA
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LinearRecurrence[{4, 2}, {1, 5}, 30] (* Amiram Eldar, Nov 26 2018 *)
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PROG
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(PARI) Vec((1 + x)/(1 - 4*x - 2*x^2) + O(x^30)) \\ Andrew Howroyd, Nov 25 2018
(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
(Sage) s=((1+x)/(1-4*x-2*x^2)).series(x, 50); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 29 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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