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A300451
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a(n) = (3*n^2 - 3*n + 8)*2^(n - 3).
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2
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1, 2, 7, 26, 88, 272, 784, 2144, 5632, 14336, 35584, 86528, 206848, 487424, 1134592, 2613248, 5963776, 13500416, 30343168, 67764224, 150470656, 332398592, 730857472, 1600126976, 3489660928, 7583301632, 16424894464, 35467034624, 76369887232, 164014063616
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OFFSET
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0,2
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COMMENTS
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1 and 7 are the only odd terms.
a(n) gives the number of words of length n defined over the alphabet {a,b,c,d} such that letters from {a,b} are only used in pairs of at most one, and consist of (a,a), (a,b) and (b,a).
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REFERENCES
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Robert A. Beeler, How to Count: An Introduction to Combinatorics and Its Applications, Springer International Publishing, 2015.
Ian F. Blake, The Mathematical Theory of Coding, Academic Press, 1975.
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LINKS
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FORMULA
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G.f.: (1 - 4*x + 7*x^2)/(1 - 6*x + 12*x^2 - 8*x^3).
E.g.f: (1/2)*(3*x^2 + 2)*exp(2*x).
a(n) = ((3/4)*binomial(n, 2) + 1)*2^n.
a(n) = 2*a(n-1) + 3*(n - 1)*2^(n - 2), with a(0) = 1.
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3), for n >= 3, with a(0) = 1, a(1) = 2 and a(2) = 7.
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EXAMPLE
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a(4) = 88. The corresponding words are cccc, cccd, ccdc, ccdd, cdcc, cdcd, cddc, cddd, dccc, dccd, dcdc, dcdd, ddcc, ddcd, dddc, dddd, caac, caca, ccaa, caad, cada, caad, cabc, cacb, ccab, cabd, cadb, cabd, cbac, cbca, ccba, cbad, cbda, cbad, daac, daca, dcaa, daad, dada, daad, dabc, dacb, dcab, dabd, dadb, dabd, dbac, dbca, dcba, dbad, dbda, dbad, aacc, acac, acca, aacd, acad, acda, aadc, adac, adca, aadd, adad, adda, abcc, acbc, accb, abcd, acbd, acdb, abdc, adbc, adcb, abdd, adbd, addb, bacc, bcac, bcca, bacd, bcad, bcda, badc, bdac, bdca, badd, bdad, bdda.
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MAPLE
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A := n -> (3*n^2 - 3*n + 8)*2^(n - 3);
seq(A(n), n = 0 .. 70);
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MATHEMATICA
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Table[(3 n^2 - 3 n + 8) 2^(n - 3), {n, 0, 70}]
CoefficientList[Series[(1 - 4x + 7x^2)/(1 - 2x)^3, {x, 0, 30}], x] (* or *)
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PROG
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(Maxima) makelist((3*n^2 - 3*n + 8)*2^(n - 3), n, 0, 70);
(PARI) a(n) = (3*n^2-3*n+8)*2^(n-3); \\ Altug Alkan, Mar 09 2018
(GAP) List([0..30], n->(3*n^2-3*n+8)*2^(n-3)); # Muniru A Asiru, Mar 09 2018
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CROSSREFS
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Cf. A000079, A000292, A001788, A005448, A006003, A045943, A052481, A053730, A081908, A295288, A300184.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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