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A048876
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a(n) = 4*a(n-1) + a(n-2); a(0)=1, a(1)=7.
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14
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1, 7, 29, 123, 521, 2207, 9349, 39603, 167761, 710647, 3010349, 12752043, 54018521, 228826127, 969323029, 4106118243, 17393796001, 73681302247, 312119004989, 1322157322203, 5600748293801, 23725150497407, 100501350283429, 425730551631123, 1803423556807921
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OFFSET
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0,2
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COMMENTS
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Generalized Pell equation with second term of 7.
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LINKS
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L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Pellian Representations, Fib. Quart. Vol. 10, No. 5, (1972), pp. 449-488.
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FORMULA
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a(n) = ((1+sqrt(5))*(2+sqrt(5))^n + (1-sqrt(5))*(2-sqrt(5))^n )/2.
a(n) = Fibonacci(3*n+7) mod Fibonacci(3*n+3), n > 0.
a(n) = Fibonacci(3*n+3) - Fibonacci(3*n-1). (End)
a(n) = 5*F(2*n)*F(n+1) - L(n-1)*(-1)^n. - J. M. Bergot, Mar 22 2016
a(n) = Sum_{k=0..n} binomial(n,k)*5^floor((k+1)/2)*2^(n-k). - Tony Foster III, Sep 03 2017
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MAPLE
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with(combinat): a:=n->3*fibonacci(n-1, 4)+fibonacci(n, 4): seq(a(n), n=1..16); # Zerinvary Lajos, Apr 04 2008
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MATHEMATICA
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f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(2 + s)^n + (1 - s)(2 - s)^n)/2]]; (* Or *)
f[n_] := Fibonacci[3 n + 3] - Fibonacci[3 n - 1]; (* Or *)
f[n_] := Mod[ Fibonacci[3n + 7], Fibonacci[3n + 3]]; Array[f, 22, 0]
a[n_] := 4a[n - 1] + a[n - 2]; a[0] = 1; a[1] = 7; Array[a, 22, 0] (* Or *)
LinearRecurrence[{4, 1}, {1, 7}, 30] (* Harvey P. Dale, Jun 13 2015 *)
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PROG
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(PARI) Vec((1+3*x)/(1-4*x-x^2) + O(x^30)) \\ Altug Alkan, Oct 07 2015
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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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