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 A229350 Decimal expansion of continued fraction [x(1),x(2),x(3),...], where x(n) = F(n+1)/F(n), where F = A000045 (Fibonacci numbers). 11
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS FORMULA [1, [1,1], [1,1,1], [1,1,1,1], ... ]. (Here, as in the Name and Example sections, square brackets indicate continued fractions.) EXAMPLE [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. MATHEMATICA 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 *) CROSSREFS Cf. A229348, A229349, A229351. Sequence in context: A200240 A199052 A021255 * A070342 A125125 A021719 Adjacent sequences: A229347 A229348 A229349 * A229351 A229352 A229353 KEYWORD nonn,cons,easy AUTHOR Clark Kimberling, Sep 21 2013 STATUS approved

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Last modified March 25 00:24 EDT 2023. Contains 361511 sequences. (Running on oeis4.)