1,2

Number of digits in Zeckendorf-binary representation of n. E.g. the Zeckendorf representation of 12 is 8+3+1, which in binary notation is 10101, which consists of 5 digits. - Clark Kimberling (ck6(AT)evansville.edu), Jun 05 2004

First position where value n occurs is A000045(n+1), i.e. a(A000045(n)) = n-1, for n >= 2 and a(A000045(n)-1) = n-2, for n >= 3.

This is the number of distinct Fibonacci numbers greater than 0 which are less than or equal to n. - Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 10 2006

The smallest non-decreasing sequence a(n) such that a(Fibonacci(n-1)) = n. - Tanya Khovanova (tanyakh(AT)yahoo.com), Jun 20 2007

G.f.: (Sum_{n>1} x^fibonacci(n))/(1-x). - Michael Somos, Apr 25, 2003

a(n)=floor(log_phi((sqr(5)*n+sqr(5*n^2+4))/2))-1, n>=1, where phi is the golden ratio. Alternatively, a(n)=floor(arsinh(sqr(5)*n/2)/ln(phi))-1. Also a(n)=A108852(n)-2. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 02 2007

a(n) = -1 + floor[ log_phi( (n+0.2)*sqrt(5) ) ], where log_phi(x) is the logarithm to the base (1+sqrt(5))/2. - Ralf Stephan, May 14 2007

1, 1, then F(2) 2's, then F(3) 3's, then F(4) 4's, ..., then F(k) k's, ...

Table[Table[n, {Fibonacci[n]}], {n, 10}] // Flatten (* Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 14 2007 *)

(PARI) a(n) = -1+floor(log(((n+0.2)*sqrt(5)))/log((1+sqrt(5))/2))

(PARI) a(n)=local(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2)

(MIT Scheme) (define (A072649 n) (let ((b (A072648 n))) (+ -1 b (floor->exact (/ n (A000045 (1+ b)))))))

Cf. A000045, A095791, A095792.

Used to construct A003714. Cf. also A002024, A072643, A072648, A072650.

Cf. A131234.

nonn,new

Antti Karttunen Jun 02 2002

Typo fixed by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Oct 28 2009