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A130234
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Minimal index k of a Fibonacci number such that Fibonacci(k) >= n (the 'upper' Fibonacci Inverse).
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23
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0, 1, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11
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OFFSET
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0,3
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COMMENTS
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Inverse of the Fibonacci sequence (A000045), nearly, since a(Fibonacci(n)) = n except for n = 2 (see A130233 for another version). a(n+1) is equal to the partial sum of the Fibonacci indicator sequence (see A104162).
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LINKS
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FORMULA
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a(n) = ceiling(log_phi((sqrt(5)*n + sqrt(5*n^2-4))/2)) = ceiling(arccosh(sqrt(5)*n/2)/log(phi)) where phi = (1+sqrt(5))/2, the golden ratio, for n > 0.
G.f.: x/(1-x) * Sum_{k >= 0} x^Fibonacci(k).
a(n) = ceiling(log_phi(sqrt(5)*n - 1)) for n > 0, where phi is the golden ratio. - Hieronymus Fischer, Jul 02 2007
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EXAMPLE
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a(10) = 7, since Fibonacci(7) = 13 >= 10 but Fibonacci(6) = 8 < 10.
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MAPLE
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local i;
for i from 0 do
return i;
end if;
end do:
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MATHEMATICA
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a[n_] := For[i = 0, True, i++, If[Fibonacci[i] >= n, Return[i]]];
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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