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Numerators of convergents to the square root of the golden ratio.
7

%I #22 Jan 05 2025 19:51:40

%S 1,4,5,14,159,491,2614,3105,33664,36769,107202,572779,1825539,9700474,

%T 11526013,32752500,142536013,175288513,317824526,810937565,3561574786,

%U 182451251651,186012826437,926502557399,2039017941235,5004538439869,157179709577174

%N Numerators of convergents to the square root of the golden ratio.

%H I. J. Good, <a href="https://doi.org/10.1057/jors.1992.123">Complex Fibonacci and Lucas Numbers, Continued Fractions, and the Square Root of the Golden Ratio (Condensed Version)</a>, Journal of the Operational Research Society, 43 (1992), 837-842.

%H I. J. Good, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/31-1/good.pdf">Complex Fibonacci and Lucas Numbers, Continued Fractions, and the Square Root of the Golden Ratio</a>, The Fibonacci Quarterly 31.1 (1993):7-20.

%F a(n) = A331692(n)*a(n-1) + a(n-2) for n >= 2. - _Jianing Song_, Aug 18 2022

%e 1, 4/3, 5/4, 14/11, 159/125, 491/386, 2614/2055, 3105/2441, ... = A225204/A225205

%t Numerator[Convergents[Sqrt[GoldenRatio], 20]]

%Y Cf. A001622, A139339, A331692, A225205 (denominators).

%K nonn,frac

%O 0,2

%A _Eric W. Weisstein_, May 01 2013