A264080 a(n) = 6*F(n)*F(n+1) + (-1)^n, where F = A000045.
1, 5, 13, 35, 91, 239, 625, 1637, 4285, 11219, 29371, 76895, 201313, 527045, 1379821, 3612419, 9457435, 24759887, 64822225, 169706789, 444298141, 1163187635, 3045264763, 7972606655, 20872555201, 54645058949, 143062621645, 374542805987, 980565796315
Offset: 0
Links
- Bruno Berselli, Table of n, a(n) for n = 0..500
- Index entries for linear recurrences with constant coefficients, signature (2,2,-1).
Crossrefs
Programs
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Magma
[6*Fibonacci(n)*Fibonacci(n+1)+(-1)^n: n in [0..30]];
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Maple
a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1,5,13>>)[1, 1]: seq(a(n), n=0..30); # Alois P. Heinz, Sep 28 2016
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Mathematica
Table[6 Fibonacci[n] Fibonacci[n + 1] + (-1)^n, {n, 0, 30}] LinearRecurrence[{2,2,-1},{1,5,13},30] (* Harvey P. Dale, Jul 12 2019 *) -
Maxima
makelist(6*fib(n)*fib(n+1)+(-1)^n, n, 0, 30);
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PARI
for(n=0, 30, print1(6*fibonacci(n)*fibonacci(n+1)+(-1)^n", "));
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PARI
a(n) = round((2^(-n)*(-(-2)^n-3*(3-sqrt(5))^n*(-1+sqrt(5))+3*(1+sqrt(5))*(3+sqrt(5))^n))/5) \\ Colin Barker, Sep 28 2016
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PARI
Vec((1+3*x+x^2)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 28 2016
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Sage
[6*fibonacci(n)*fibonacci(n+1)+(-1)^n for n in (0..30)]
Formula
G.f.: (1+3*x+x^2) / ((1+x)*(1-3*x+x^2)). - Corrected by Colin Barker, Sep 28 2016
a(n) = -a(-n-1) = 2*a(n-1) + 2*a(n-2) - a(n-3) for all n in Z.
a(n) - a(n-1) = 2*A099016(n) with a(-1)=-1.
a(n) + a(n-1) = 2*A097134(n) for n>0.
Sum_{i>=0} 1/a(i) = 1.3232560865206157372628688449331...
a(n) = (2^(-n)*(-(-2)^n-3*(3-sqrt(5))^n*(-1+sqrt(5))+3*(1+sqrt(5))*(3+sqrt(5))^n))/5. - Colin Barker, Sep 28 2016
E.g.f.: (1/5)*exp(-x)*(-1 + 6*exp(5*x/2)*(cosh((sqrt(5)*x)/2) + sqrt(5)*sinh((sqrt(5)*x)/2))). - Stefano Spezia, Dec 09 2019
Comments