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A229350
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Decimal expansion of continued fraction [x(1),x(2),x(3),...], where x(n) = F(n+1)/F(n), where F = A000045 (Fibonacci numbers).
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11
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1, 3, 9, 8, 5, 9, 8, 5, 1, 6, 3, 2, 9, 3, 7, 8, 7, 1, 8, 7, 8, 5, 9, 0, 5, 5, 5, 2, 8, 1, 7, 6, 1, 4, 1, 8, 5, 5, 6, 0, 3, 3, 6, 5, 4, 0, 9, 8, 8, 4, 6, 3, 9, 3, 6, 4, 5, 0, 0, 9, 2, 0, 2, 4, 8, 8, 5, 5, 6, 0, 9, 7, 1, 6, 4, 7, 0, 4, 8, 1, 5, 5, 6, 8, 8, 1
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OFFSET
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1,2
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COMMENTS
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Suppose that x(n) is a sequence of positive real numbers with divergent sum. By the Seidel Convergence Theorem, the continued fraction [x(1),x(2),x(3),...] converges.
The fact that F(n+1)/F(n) is the n-th convergent of a continued fraction (specifically, of the golden ratio) exemplifies a certain function f of a positive real variable x: let p(i)/q(i), for i >=0, be the convergents to x; then f(x) is the number [p(0)/q(0), p(1)/q(1), p(2)/q(2), ... ]. For x = golden ratio, f(x) = 1.398598..., f(f(x)) = 1.4903397..., f(f(f(x))) = 1.4995061.... Let L(x) = lim(f(n,x)), where f(0,x) = x, f(1,x) = f(x), and f(n,x) = f(f(n-1,x)). It appears that L(golden ratio) = 3/2.
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LINKS
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Table of n, a(n) for n=1..86.
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FORMULA
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[1, [1,1], [1,1,1], [1,1,1,1], ... ]. (Here, as in the Name and Example sections, square brackets indicate continued fractions.)
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EXAMPLE
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[1, 2/1, 3/2, 5/3, 8/5,...] = [1,2,1,1,27,1,16,670,9,3,2,1,13,1,4,1,1,1...] = 1.3985985... The first 5 ordinary convergents are 1, 3/2, 4/3, 7/5, 193/138.
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MATHEMATICA
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z = 500; t = Table[Fibonacci[n + 1]/Fibonacci[n], {n, z}]
r = FromContinuedFraction[t]; c = Convergents[r, z];
Numerator[c] (* A229348 *)
Denominator[c] (* A229349 *)
RealDigits[r, 10, 120] (* A229350 *)
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CROSSREFS
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Cf. A229348, A229349, A229351.
Sequence in context: A200240 A199052 A021255 * A070342 A125125 A021719
Adjacent sequences: A229347 A229348 A229349 * A229351 A229352 A229353
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KEYWORD
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nonn,cons,easy
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AUTHOR
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Clark Kimberling, Sep 21 2013
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STATUS
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approved
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