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A179057 a(n) is the smallest argument m for which an auxiliary sequence A_n(m) differs from Fibonacci(m). 1
9, 9, 13, 19, 23, 29, 33, 42 (list; graph; refs; listen; history; text; internal format)



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, 1, 2, 3,...

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.


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


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.


Cf. A000045

Sequence in context: A118662 A175219 A124475 * A144418 A003885 A168390

Adjacent sequences:  A179054 A179055 A179056 * A179058 A179059 A179060




Vladimir Shevelev, Jun 27 2010


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



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Last modified May 25 16:15 EDT 2017. Contains 287039 sequences.