A063726 a(n) = gcd(1 + Fibonacci(n+1), 1 + Fibonacci(n)).

1, 2, 1, 1, 2, 3, 1, 2, 1, 7, 2, 5, 1, 18, 1, 13, 2, 47, 1, 34, 1, 123, 2, 89, 1, 322, 1, 233, 2, 843, 1, 610, 1, 2207, 2, 1597, 1, 5778, 1, 4181, 2, 15127, 1, 10946, 1, 39603, 2, 28657, 1, 103682, 1, 75025, 2, 271443, 1, 196418, 1, 710647, 2, 514229, 1, 1860498, 1

Offset: 0

Author

Jason Earls, Aug 11 2001

Keywords

• nonn

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000045, A001611.

Programs

GAP

List([0..65],n->Gcd(1+Fibonacci(n+1),1+Fibonacci(n))); # Muniru A Asiru, Oct 09 2018

Magma

[GCD(1 + Fibonacci(n+1), 1 + Fibonacci(n)): n in [0..50]]; // G. C. Greubel, Oct 08 2018

Maple

with(combinat): seq(gcd(1+fibonacci(n+1),1+fibonacci(n)),n=0..65); # Muniru A Asiru, Oct 09 2018

Mathematica

Table[GCD[Fibonacci[n],Fibonacci[n+1]+1],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2010 *)

PARI

j=[]; for(n=0,75,j=concat(j,gcd(1+fibonacci(n+1),1+fibonacci(n) ))); j

PARI

{ g=0; f=1; for (n=0, 1000, write("b063726.txt", n, " ", gcd(1 + f, 1 + g)); h=g; g=f; f+=h ) } \\ Harry J. Smith, Aug 28 2009

Formula

Conjectures from Colin Barker, Jan 30 2018: (Start)

G.f.: (1 + 2*x + x^2 + x^3 - x^4 - 3*x^5 - 3*x^6 - 3*x^7 - 5*x^8 - x^9 + x^10 + 3*x^11 + 2*x^12 + x^13) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2)*(1 + x^2 - x^4)*(1 - x^2 - x^4)).

a(n) = 3*a(n-4) + a(n-6) - a(n-8) - 3*a(n-10) + a(n-14) for n>13.

(End)