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A179203 The smallest argument m for which an approximating sequence B_n(m) differs from Fibonacci(m). 1
3, 10, 12, 23, 23, 28, 30, 35, 40, 45, 51, 54, 59, 64, 70, 74, 80, 83, 91, 99, 99 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Given n, an auxiliary sequence B_n(m) is defined by B_n(m) = A000045(m), 0 <= m < 3 and B_n(m) = round(x_n*B_n(m-1)), m >= 3, where x_n is a truncated approximation of the golden ratio A001622 = 1.61803398..., namely, x_n = floor(A001622*10^n)/10^n = 1, 1.6, 1.61, 1.618, ... If one were to replace x_n with the exact value of golden ratio, the B_n(m) would reproduce the Fibonacci sequence. The sequence shows the index where B_n(m) diverges first from Fibonacci(m): B_n(m) = Fibonacci(m) for 0 <= m < a(n) and B_n(m) < Fibonacci(m) for m=a(n).

LINKS

Table of n, a(n) for n=0..20.

EXAMPLE

For n=1 and m>=3, we have B_1(m) = round(1.6*B_(m-1)).By this formula with the initial conditions, B_1(3)=2, B_1(4)=3, B_1(5)=5, B_1(6)=8, B_1(7)=13, B_1(8)=21, B_1(9)=34 and B_1(10)=54. Since F(10)=55, then B_1(m) gives the first 10 Fibonacci numbers: F(0),...,F(9). Thus a(1)=10.

MAPLE

A179203 := proc(n)local a001622, x, B ; a001622 := (1+sqrt(5))/2 ; x := floor( a001622*10^n)/10^n ; B := combinat[fibonacci](2) ;

for m from 3 do B := round(x*B) ; if B <> combinat[fibonacci](m) then return m; end if; end do:

end proc:

seq(A179203(n), n=0..20) ; # R. J. Mathar, Jan 04 2011

CROSSREFS

Cf. A000045, A001622, A179057.

Sequence in context: A317671 A031453 A345961 * A102017 A032916 A044994

Adjacent sequences:  A179200 A179201 A179202 * A179204 A179205 A179206

KEYWORD

nonn,base,less

AUTHOR

Vladimir Shevelev, Jul 02 2010

EXTENSIONS

a(8), a(9) corrected, sequence extended by R. J. Mathar, Jan 04 2011

STATUS

approved

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Last modified August 4 13:49 EDT 2021. Contains 346447 sequences. (Running on oeis4.)