login
Indicator sequence for the Fibonacci numbers.
18

%I #27 Nov 04 2024 12:49:19

%S 1,2,1,1,0,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,

%T 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1

%N Indicator sequence for the Fibonacci numbers.

%C Without multiplicities, this is A010056.

%C The number of nonnegative integer solutions of x^4 - 10*n^2*x^2 + 25*n^4 - 16 = 0. - _Hieronymus Fischer_, May 17 2007

%F G.f.: Sum_{k>=0} x^Fibonacci(k).

%F From _Hieronymus Fischer_, May 17 2007: (Start)

%F a(n) = 1+floor(arcsinh(sqrt(5)*n/2)/log(phi))-ceiling(arccosh(sqrt(5)*n/2)/log(phi)), for n>0, where phi=(1+sqrt(5))/2.

%F a(n) = A108852(n) - A108852(n-1) for n>0.

%F a(n) = A130233(n) - A130233(n-1) for n>0.

%F a(n) = 1 + A130233(n) - A130234(n) for n>0.

%F a(n) = A130234(n+1) - A130234(n) for n>=0. (End)

%e a(1)=2 since F(1)=F(2)=1.

%o (PARI) a(n)=if(n==1,return(2)); my(k=n^2);k+=((k + 1) << 2);issquare(k) || issquare(k-8) \\ _Charles R Greathouse IV_, Feb 03 2014; typo corrected by _Georg Fischer_, Jun 22 2022

%Y Cf. A000045.

%Y Partial sums are in A108852.

%Y See also A130233 and A130234.

%K easy,nonn,changed

%O 0,2

%A _Paul Barry_, Apr 01 2005