A260259 a(n) = F(n)*F(n+1) - (-1)^n, where F = A000045.

-1, 2, 1, 7, 14, 41, 103, 274, 713, 1871, 4894, 12817, 33551, 87842, 229969, 602071, 1576238, 4126649, 10803703, 28284466, 74049689, 193864607, 507544126, 1328767777, 3478759199, 9107509826, 23843770273, 62423800999, 163427632718, 427859097161, 1120149658759

Offset: 0

Author

Bruno Berselli, Oct 31 2015

Keywords

• sign
• easy

Comments

Primes in sequence for n = 1, 3, 5, 6, 9, 24, 42, 48, 53, 71, 86, 102, 138, 182, 302, 438, 506, 926, ...

Links

• Bruno Berselli, Table of n, a(n) for n = 0..500
• A. Bremner, R. Høibakk, D. Lukkassen, Crossed ladders and Euler’s quartic, Annales Mathematicae et Informaticae, 36 (2009) pp. 29-41. See p. 33.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Crossrefs

First bisection of A111569.

Cf. A226205: numbers of the form F(n)*F(n+1)+(-1)^n.

Cf. A000045, A001654, A003482, A059929, A089508 (first bisection, without -1), A206351.

Programs

Magma

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

Maple

with(combinat): A260259:=n->fibonacci(n)*fibonacci(n+1)-(-1)^n: seq(A260259(n), n=0..50); # Wesley Ivan Hurt, Feb 04 2017

Mathematica

Table[Fibonacci[n] Fibonacci[n + 1] - (-1)^n, {n, 0, 30}]

Maxima

makelist(fib(n)*fib(n+1)-(-1)^n,n,0,30);

PARI

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

PARI

a(n) = round((2^(-1-n)*(-3*(-1)^n*2^(2+n)-(3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n))/5) \\ Colin Barker, Sep 29 2016

PARI

Vec(-(1-4*x+x^2)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 29 2016

Sage

[fibonacci(n)*fibonacci(n+1)-(-1)^n for n in (0..30)]

Formula

G.f.: (-1 + 4*x - x^2)/((1 + x)*(1 - 3*x + x^2)).

a(n) = -a(-n-1) = 2*a(n-1) + 2*a(n-2) - a(n-3) for all n in Z.

a(n) = F(n+2)^2 - 2*F(n+1)^2.

a(n) = A059929(n) - A059929(n-1) with A059929(-1)=1.

a(n) = -A001654(n+1) + 4*A001654(n) - A001654(n-1).

a(n) = A206351((n+2)/2)-2 for even n; a(n) = A003482((n-1)/2)+2 for odd n.

Sum_{i>=0} 1/a(i) = .754301907697893871765121109686...

a(n) = (2^(-1-n)*(-3*(-1)^n*2^(2+n)-(3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n))/5. - Colin Barker, Sep 29 2016