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A110111
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Sequence associated to the recurrence b(n) = b(n-1) + 3*b(n-2).
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1
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0, 1, 7, 133, 1330, 18430, 210490, 2673223, 31940881, 394918819, 4788779380, 58709030380, 715296121540, 8745656280829, 106717441265323, 1303667366328817, 15915556720909510, 194371775990116810
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OFFSET
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0,3
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COMMENTS
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In general, let T(n,k) be the solution to T(n,k) = T(n-1,k) + k*T(n-2,k) for n >= 2 with T(0,k) = 0 and T(1,k) = 1 for all k. Then, for fixed k, S(n,k) = T(n,k) * T(n+1,k) * T(n+2,k)/(k+1) has g.f. x/((1 + k*x - k^3*x^2) * (1 - (3*k + 1)*x - k^3*x^2)) (cf. A110112). For the current sequence, a(n) = S(n,k=3) = T(n,3) * T(n+1,3) * T(n+2,3)/4. [Edited by Petros Hadjicostas, Dec 26 2019]
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LINKS
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FORMULA
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G.f.: x/((1 + 3*x - 27*x^2) * (1 - 10*x - 27*x^2)).
a(n) = 7*a(n-1) + 84*a(n-2) - 189*a(n-3) - 729*a(n-4) for n >= 4.
a(n) = b(n) * b(n+1) * b(n+2)/4, where b(n) = (((1 + sqrt(13))/2 )^n - ((1 - sqrt(13))/2)^n)/sqrt(13). [Corrected by Petros Hadjicostas, Dec 26 2019]
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MATHEMATICA
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CoefficientList[Series[x/((1+3x-27x^2)(1-10x-27x^2)), {x, 0, 30}], x] (* or *) LinearRecurrence[{7, 84, -189, -729}, {0, 1, 7, 133}, 30] (* Harvey P. Dale, Jun 29 2022 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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