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%I #8 Sep 24 2013 12:27:29
%S 1,2,3,5,138,143,2426,1625563,14632493,45523042,105678577,151201619,
%T 2071299624,2222501243,10961304596,13183805839,24145110435,
%U 37328916274,248118608079,285447524353,3388041375962,7061530276277,10449571652239,27960673580755
%N Denominators of the ordinary convergents of continued fraction [x(1),x(2),x(3),...], where x(n) = F(n+1)/F(n), where 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. A229348, A229350, A229351.
%K nonn,frac,easy
%O 1,2
%A _Clark Kimberling_, Sep 21 2013
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