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A229348 Numerators of the ordinary convergents of continued fraction [x(1),x(2),x(3),...], where x(n) = F(n+1)/F(n), F = A000045 (Fibonacci numbers). 6

%I #8 Sep 24 2013 12:27:02

%S 1,3,4,7,193,200,3393,2273510,20464983,63668459,147801901,211470360,

%T 2896916581,3108386941,15330464345,18438851286,33769315631,

%U 52208166917,347018317133,399226484050,4738509641683,9876245767416,14614755409099,39105756585614

%N Numerators of the ordinary convergents of continued fraction [x(1),x(2),x(3),...], where x(n) = F(n+1)/F(n), F = A000045 (Fibonacci numbers).

%C 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.

%e [1, 2/1, 3/2, 5/3, 8/5,...] = 1.3985985... The first 5 ordinary convergents are 1, 3/2, 4/3, 7/5, 193/138.

%t z = 500; t = Table[Fibonacci[n + 1]/Fibonacci[n], {n, z}]

%t r = FromContinuedFraction[t]; c = Convergents[r, z];

%t Numerator[c] (* A229348 *)

%t Denominator[c] (* A229349 *)

%t RealDigits[r, 10, 120] (* A229350 *)

%Y Cf. A229349, A229350, A229351.

%K nonn,frac,easy

%O 1,2

%A _Clark Kimberling_, Sep 21 2013

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Last modified May 24 22:09 EDT 2024. Contains 372782 sequences. (Running on oeis4.)