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A179203
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The smallest argument m for which an approximating sequence B_n(m) differs from Fibonacci(m).
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1
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3, 10, 12, 23, 23, 28, 30, 35, 40, 45, 51, 54, 59, 64, 70, 74, 80, 83, 91, 99, 99
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OFFSET
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0,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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CROSSREFS
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KEYWORD
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nonn,base,less
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AUTHOR
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EXTENSIONS
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a(8), a(9) corrected, sequence extended by R. J. Mathar, Jan 04 2011
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STATUS
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approved
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