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A300451 a(n) = (3*n^2 - 3*n + 8)*2^(n - 3).

%I A300451 #46 Sep 08 2022 08:46:20
%S A300451 1,2,7,26,88,272,784,2144,5632,14336,35584,86528,206848,487424,
%T A300451 1134592,2613248,5963776,13500416,30343168,67764224,150470656,
%U A300451 332398592,730857472,1600126976,3489660928,7583301632,16424894464,35467034624,76369887232,164014063616
%N A300451 a(n) = (3*n^2 - 3*n + 8)*2^(n - 3).
%C A300451 First difference yields A295288.
%C A300451 1 and 7 are the only odd terms.
%C A300451 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).
%D A300451 Robert A. Beeler, How to Count: An Introduction to Combinatorics and Its Applications, Springer International Publishing, 2015.
%D A300451 Ian F. Blake, The Mathematical Theory of Coding, Academic Press, 1975.
%H A300451 G. C. Greubel, <a href="/A300451/b300451.txt">Table of n, a(n) for n = 0..1000</a>
%H A300451 Hermann Gruber, Jonathan Lee and Jeffrey Shallit, <a href="https://arxiv.org/abs/1204.4982">Enumerating regular expressions and their languages</a>, arXiv preprint arXiv:1204.4982 [cs.FL], 2012.
%H A300451 Luis Manuel Rivera, <a href="https://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
%H A300451 Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10569">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.
%H A300451 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%F A300451 G.f.:  (1 - 4*x + 7*x^2)/(1 - 6*x + 12*x^2 - 8*x^3).
%F A300451 E.g.f: (1/2)*(3*x^2 + 2)*exp(2*x).
%F A300451 a(n) = ((3/4)*binomial(n, 2) + 1)*2^n.
%F A300451 a(n) = 2*a(n-1) + 3*(n - 1)*2^(n - 2), with a(0) = 1.
%F A300451 a(n) = 3*A001788(n) + A000079(n).
%F A300451 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.
%F A300451 a(n) = A300184(n,2).
%e A300451 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.
%p A300451 A := n -> (3*n^2 - 3*n + 8)*2^(n - 3);
%p A300451 seq(A(n), n = 0 .. 70);
%t A300451 Table[(3 n^2 - 3 n + 8) 2^(n - 3), {n, 0, 70}]
%t A300451 CoefficientList[Series[(1 - 4x + 7x^2)/(1 - 2x)^3, {x, 0, 30}], x] (* or *)
%t A300451 LinearRecurrence[{6, -12, 8}, {1, 2, 7}, 30] (* _Robert G. Wilson v_, Mar 07 2018 *)
%o A300451 (Maxima) makelist((3*n^2 - 3*n + 8)*2^(n - 3), n, 0, 70);
%o A300451 (PARI) a(n) = (3*n^2-3*n+8)*2^(n-3); \\ _Altug Alkan_, Mar 09 2018
%o A300451 (GAP) List([0..30],n->(3*n^2-3*n+8)*2^(n-3)); # _Muniru A Asiru_, Mar 09 2018
%o A300451 (Magma) [(3*n^2-3*n+8)*2^(n-3): n in [0..30]]; // _Vincenzo Librandi_, Mar 10 2018
%Y A300451 Cf. A000079, A000292, A001788, A005448, A006003, A045943, A052481, A053730, A081908, A295288, A300184.
%K A300451 nonn,easy
%O A300451 0,2
%A A300451 _Franck Maminirina Ramaharo_, Mar 06 2018