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Decimal expansion of Sum_{k>=1} H(k)*L(k)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and L(k) = A000032(k) is the k-th Lucas number.
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%I #10 Jan 05 2025 19:51:42

%S 8,4,6,2,9,7,2,4,9,2,9,9,9,7,1,2,2,4,5,3,9,7,7,2,5,0,5,8,2,5,5,1,1,3,

%T 6,6,2,6,9,8,7,0,7,6,3,1,5,6,4,4,2,8,0,7,2,2,9,4,1,4,1,0,9,6,8,8,5,9,

%U 7,3,8,8,6,4,2,9,4,8,7,9,0,7,2,5,0,0,8,2,6,0,8,9,5,0,7,1,1,6,7,9,3,1,5,3,1

%N Decimal expansion of Sum_{k>=1} H(k)*L(k)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and L(k) = A000032(k) is the k-th Lucas number.

%H Hideyuki Ohtsuka, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbSolnNov2016.pdf">Problem B-1200</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 54, No. 4 (2016), p. 367; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbSolnNov2017.pdf">Harmonic and Fiboancci [sic]/Lucas Numbers</a>, Solution to Problem B-1200 by Kenny B. Davenport, ibid., Vol. 55, No. 4 (2017), pp. 372-373.

%F Equals log(64*phi^(4*sqrt(5))) = 6*log(2) + 4*sqrt(5)*log(phi), where phi is the golden ratio (A001622).

%e 8.46297249299971224539772505825511366269870763156442...

%t RealDigits[6*Log[2] + 4*Sqrt[5]*Log[GoldenRatio], 10, 100][[1]]

%Y Cf. A000032, A001008, A001622, A002162, A002390, A002805, A349850.

%K nonn,cons

%O 1,1

%A _Amiram Eldar_, Dec 02 2021