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a(n) = gcd(1 + Fibonacci(n+1), 1 + Fibonacci(n)).
3

%I #27 Sep 08 2022 08:45:04

%S 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,

%T 843,1,610,1,2207,2,1597,1,5778,1,4181,2,15127,1,10946,1,39603,2,

%U 28657,1,103682,1,75025,2,271443,1,196418,1,710647,2,514229,1,1860498,1

%N a(n) = gcd(1 + Fibonacci(n+1), 1 + Fibonacci(n)).

%H Harry J. Smith, <a href="/A063726/b063726.txt">Table of n, a(n) for n = 0..1000</a>

%F Conjectures from _Colin Barker_, Jan 30 2018: (Start)

%F 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)).

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

%F (End)

%p with(combinat): seq(gcd(1+fibonacci(n+1),1+fibonacci(n)),n=0..65); # _Muniru A Asiru_, Oct 09 2018

%t Table[GCD[Fibonacci[n],Fibonacci[n+1]+1],{n,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 03 2010 *)

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

%o (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

%o (Magma) [GCD(1 + Fibonacci(n+1), 1 + Fibonacci(n)): n in [0..50]]; // _G. C. Greubel_, Oct 08 2018

%o (GAP) List([0..65],n->Gcd(1+Fibonacci(n+1),1+Fibonacci(n))); # _Muniru A Asiru_, Oct 09 2018

%Y Cf. A000045, A001611.

%K nonn

%O 0,2

%A _Jason Earls_, Aug 11 2001