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A157459
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Expansion of 72*x^2 / (1 - 323*x + 323*x^2 - x^3).
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4
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0, 72, 23256, 7488432, 2411251920, 776415629880, 250003421569512, 80500325329753056, 25920854752758914592, 8346434730063040745640, 2687526062225546361181560, 865375045601895865259716752, 278648077157748243067267612656, 89723815469749332371794911558552
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OFFSET
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1,2
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COMMENTS
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This sequence is part of a solution of a more general problem involving two equations, three sequences a(n), b(n), c(n) and a constant A:
A * c(n) + 1 = a(n)^2,
(A+1) * c(n) + 1 = b(n)^2; for details see comment in A157014.
A157459 is the c(n) sequence for A=4.
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LINKS
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FORMULA
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G.f.: 72*x^2/(1 - 323*x + 323*x^2 - x^3).
c(1) = 0, c(2) = 72, c(3) = 323*c(2), c(n) = 323*(c(n-1) - c(n-2)) + c(n-3) for n>3.
a(n) = -((161+72*sqrt(5))^(-n)*(-1+(161+72*sqrt(5))^n)*(9+4*sqrt(5)+(-9+4*sqrt(5))*(161+72*sqrt(5))^n))/80. - Colin Barker, Jul 25 2016
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MATHEMATICA
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LinearRecurrence[{323, -323, 1}, {0, 72, 23256}, 20] (* Harvey P. Dale, Feb 28 2021 *)
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PROG
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(PARI) a(n) = -round((161+72*sqrt(5))^(-n)*(-1+(161+72*sqrt(5))^n)*(9+4*sqrt(5)+(-9+4*sqrt(5))*(161+72*sqrt(5))^n))/80 \\ Colin Barker, Jul 25 2016
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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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