A349850 -id:A349850 - cp's OEIS frontend

A349851 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.

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, 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, 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

Offset: 1

Amiram Eldar, Dec 02 2021

			8.46297249299971224539772505825511366269870763156442...

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.

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

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

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

A351789 Decimal expansion of Sum_{k>=1} AH(k)*F(k)/2^k, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 19 2022

			1.51437037420622187243459478916165077964831313316887...

Seán M. Stewart, Problem H-893, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 60, No. 1 (2022), p. 91.

Cf. A000045, A001622, A002390, A058312, A058313, A349850, A351794.

RealDigits[Log[5/4] + 6*Log[GoldenRatio]/Sqrt[5], 10, 100][[1]]

Equals log(5/4) + 6*log(phi)/sqrt(5), where phi is the golden ratio (A001622) (Stewart, 2022).

A351794 Decimal expansion of Sum_{k>=1} AH(k)*L(k)/2^k, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic number and L(k) = A000032(k) is the k-th Lucas number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 19 2022

			2.82147535876264946174605143568253063724665666934546...

Cf. A000032, A001622, A002390, A058312, A058313, A349850, A349851, A351789.

RealDigits[3*Log[5/4] + 10*Log[GoldenRatio]/Sqrt[5], 10, 100][[1]]

Equals 3*log(5/4) + 10*log(phi)/sqrt(5), where phi is the golden ratio (A001622).

Equals (4/3)*log(5/4) + (5/3)*A351789.

A370742 Decimal expansion of Sum_{k>=2} H(k-1) * F(k) / (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.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Feb 29 2024

			0.59667348783398269737770682436833083924687967042183...

Kenny B. Davenport, Problem B-1222, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 56, No. 1 (2018), p. 81; The Generating Function for Harmonic Numbers, Solution to Problem B-1222 by Amanda M. Andrews and Samantha L. Zimmerman, ibid., Vol. 57, No. 1 (2019), pp. 83-84.

Cf. A000045, A001008, A001622, A002162, A002163, A002390, A002805, A349850, A370743.

RealDigits[4 * Log[2] * Log[GoldenRatio] / Sqrt[5], 10, 120][[1]]

PARI
```
4 * log(2) * log(quadgen(5)) / sqrt(5)
```

Equals 4 * log(2) * log(phi) / sqrt(5), where phi is the golden ratio (A001622) (Davenport, 2018).

cp's OEIS Frontend

A349851 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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A351789 Decimal expansion of Sum_{k>=1} AH(k)*F(k)/2^k, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

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A351794 Decimal expansion of Sum_{k>=1} AH(k)*L(k)/2^k, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic number and L(k) = A000032(k) is the k-th Lucas number.

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A370742 Decimal expansion of Sum_{k>=2} H(k-1) * F(k) / (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.

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