A349850 Decimal expansion of Sum_{k>=1} H(k)*F(k)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.
3, 9, 6, 8, 7, 4, 8, 0, 0, 6, 9, 0, 3, 9, 1, 4, 8, 5, 2, 1, 7, 1, 0, 6, 3, 6, 4, 0, 6, 1, 9, 9, 8, 5, 6, 8, 8, 6, 9, 8, 4, 2, 4, 3, 6, 3, 9, 6, 2, 2, 4, 8, 4, 3, 6, 7, 8, 3, 3, 9, 6, 6, 4, 2, 9, 4, 2, 1, 5, 4, 5, 3, 6, 7, 0, 6, 1, 8, 1, 1, 9, 9, 3, 8, 0, 6, 6, 8, 2, 4, 2, 1, 7, 6, 1, 5, 7, 1, 0, 7, 5, 2, 1, 9, 8
Offset: 1
Examples
3.96874800690391485217106364061998568869842436396224...
Links
- Hideyuki Ohtsuka, Problem B-1200, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 54, No. 4 (2016), p. 367; Harmonic and Fiboancci [sic]/Lucas Numbers, Solution to Problem B-1200 by Kenny B. Davenport, ibid., Vol. 55, No. 4 (2017), pp. 372-373.
Programs
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Mathematica
RealDigits[2*Log[2] + 12*Log[GoldenRatio]/Sqrt[5], 10, 100][[1]]
Formula
Equals log(4*phi^(12/sqrt(5))) = 2*log(2) + 12*log(phi)/sqrt(5), where phi is the golden ratio (A001622).