A179057 a(n) is the smallest argument m for which an auxiliary sequence A_n(m) differs from Fibonacci(m).

9, 9, 13, 19, 23, 29, 33, 42

Offset

0,1

Comments

Given n, an auxiliary sequence A_n(m) is defined by A_n(m)=A000045(m), 0<=m<5 and A_n(m)=round( log_2(x_n^A_n(m-1)+x_n^A_n(m-2))), m>=5, where x_n is a truncated approximation of 2^A001622 = 3.0695645076529..., namely x_n = floor(2^A001622*10^n)/10^n = 3, 3.0, 3.06, 3.069, 3.0695,... for n >= 0.

If one would replace x_n by the exact value of 2^(golden ratio), the A_n(m) would reproduce the Fibonacci sequence.

The sequence shows the index where A_n(m) diverges first from Fibonacci(m): A_n(m) = Fibonacci(m) for 0<=m<a(n) and A_n(m) <> Fibonacci(m) for m=a(n). More exactly, it could be proved that, for m=a(n), A_n(m)=Fibonacci(m)-1.

Links

Table of n, a(n) for n=0..7.

Example

For n=0 and m>=5, we have A_0(m) = round(log_2(3^A_0(m-1)+3^A_0(m-2))). By this formula with the initial conditions, A_0(5)=5, A_0(6)=8, A_0(7)=13, A_0(8)=21 and A_0(9)=33. Since F(9)=34, then A_(m) gives the first 9 Fibonacci numbers: F(0),...,F(8). Thus a(0)=9.

Crossrefs

Cf. A001622, A000045.

Sequence in context: A305503 A175219 A124475 * A350919 A144418 A003885

Adjacent sequences: A179054 A179055 A179056 * A179058 A179059 A179060

Keyword

nonn,base,less,more

Author

Vladimir Shevelev, Jun 27 2010

Extensions

I adopted a revision of this sequence proposed by R. J. Mathar. - Vladimir Shevelev, Jun 30 2010

Status

approved