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

Author

Bruno Berselli, Nov 03 2015

Keywords

• nonn
• easy

Comments

a(n) is prime for n = 1, 2, 5, 7, 14, 15, 29, 40, 49, 57, 70, 87, 105, 127, 175, 279, 362, 647, 727, ...

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

Cf. A000032, A000045; A001654, A002878.

Cf. similar sequences of the type k*F(n)*F(n+1)+(-1)^n: A226205 (k=1); A236428 (k=2); A014742 (k=3); A061647 (k=4); A002878 (k=5).

Programs

Magma

[6*Fibonacci(n)*Fibonacci(n+1)+(-1)^n: n in [0..30]];

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

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);

PARI

for(n=0, 30, print1(6*fibonacci(n)*fibonacci(n+1)+(-1)^n", "));

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

PARI

Vec((1+3*x+x^2)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 28 2016

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) = L(2*n+1) + F(n)*F(n+1) = A002878(n) + A001654(n). See similar identity for A061647.

a(n) = A001654(n+1) + 3*A001654(n) + A001654(n-1).

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