A016627 -id:A016627 - cp's OEIS frontend

A344041 Decimal expansion of Sum_{k>=1} F(k)/(k*2^k), where F(k) is the k-th Fibonacci number (A000045).

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

Offset: 0

Amiram Eldar, May 07 2021

This constant is a transcendental number (Adhikari et al., 2001).
A similar series is Sum_{k>=1} F(k)/2^k = 2.
The corresponding series with Lucas numbers (A000032) is Sum_{k>=1} L(k)/(k*2^k) = 2*log(2) (A016627).
In general, for m>=2, Sum_{k>=1} F(k)/(k*m^k) = log(1 - 2*sqrt(5)/(1 + sqrt(5) - 2*m)) / sqrt(5) and Sum_{k>=1} L(k)/(k*m^k) = log(m^2 / (m^2 - m - 1)). - Vaclav Kotesovec, May 08 2021

			0.86081788192800807777886646590121085084914136508057...

S. D. Adhikari, N. Saradha, T. N. Shorey and R. Tijdeman, Transcendental infinite sums, Indagationes Mathematicae, Vol. 12, No. 1 (2001), pp. 1-14.
István Mező, Several Generating Functions for Second-Order Recurrence Sequences, Journal of Integer Sequences, Vol. 12 (2009), Article 09.3.7.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27.
Index entries for transcendental numbers.

Cf. A000032, A000045, A002163, A002390, A002457, A036289.

RealDigits[Sum[Fibonacci[n]/n/2^n, {n, 1, Infinity}], 10, 100][[1]]

suminf(k=1, fibonacci(k)/(k*2^k)) \\ Michel Marcus, May 07 2021

Equals Sum_{k>=0} (-1)^k/A002457(k).
Equals 4*log(phi)/sqrt(5) = 4*arcsinh(1/2)/sqrt(5) = arccosh(7/2)/sqrt(5) = 4*A002390/A002163.
Equals Integral_{x>=2} 1/(x^2 - x - 1) dx.

A352485 Decimal expansion of the probability that when a unit interval is broken at two points uniformly and independently chosen at random along its length the lengths of the resulting three intervals are the altitudes of a triangle.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 18 2022

			0.23298145831360969333463975908145302101896963809669...

Mohammed Yaseen, Table of n, a(n) for n = 0..10000
Eugen J. Ionascu, Problem 11663, The American Mathematical Monthly, Vol. 119, No. 8 (2012), pp. 699-706; alternative link; Are Random Breaks the Altitudes of a Triangle?, Solution to Problem 116633, by David Farnsworth and James Marento, ibid., Vol. 121, No. 8 (2014), pp. 741-743.
Eugen J. Ionaşcu and Gabriel Prăjitură, Things to do with a broken stick, International Journal of Geometry, Vol. 2, No. 2 (2013), pp. 5-30; arXiv preprint, arXiv:1009.0890 [math.HO], 2010-2013.
Roberto Tauraso, Problem 11663.
Eric Weisstein's World of Mathematics, Altitude.

Cf. A001622, A002390.

Similar sequences: A016627, A019702, A084623, A243398, A339392, A339393, A352484.

RealDigits[24*Sqrt[5]*Log[GoldenRatio]/25 - 4/5, 10, 100][[1]]

Equals 12*sqrt(5)*log((3+sqrt(5))/2)/25 - 4/5.

Equals 24*sqrt(5)*log(phi)/25 - 4/5, where phi is the golden ratio (A001622).

A358516 Decimal expansion of Sum_{k >= 1} (-1)^(k+1)1/((k+2)(k+3)).

Original entry on oeis.org

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

Offset: 0

Claude H. R. Dequatre, Nov 20 2022

			0.0529610277865572855011309095830198028176669353...

Michael I. Shamos, A catalog of the real numbers (2011) p.100.

Cf. A002378, A242023, A242024, A358517, A358519.

Cf. A002162.

Join[{0}, RealDigits[2*Log[2] - 4/3, 10, 120][[1]]] (* Amiram Eldar, Nov 21 2022 *)

PARI
```
2*log(2) - 4/3
```

Equals Sum_{k >= 1} (-1)^(k+1)*/((k+2)*(k+3)) = A016627 -4/3.
Equals 2*log(2) - 4/3 = Sum_{k >= 2} 1/(4*k^3 - k) = Sum_{k >= 1} (zeta(2*k + 1) - 1)/(4^k). [from the Shamos reference]
Equals Sum_{k >= 1} 1/((2^k)*(4*k + 12)). [from the Shamos reference]
Equals Sum_{k>=3} (-1)^(k+1)/A002378(k). - Amiram Eldar, Nov 21 2022

Missing terms 6, 0 inserted after a(74) by Georg Fischer, Feb 07 2025

A358646 Decimal expansion of 3/4 + log(4).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 24 2022

			2.13629436111989061883446424291635313615100026872051050824136...

Jason Bard, Table of n, a(n) for n = 1..10000
Joerg Bruedern and Trevor D. Wooley, Partitio Numerorum: sums of a prime and a number of k-th powers, arXiv:2211.10387 [math.NT], 2022, see p. 15.
Index entries for transcendental numbers.

Cf. A016627, A358645.

Mathematica
```
First[RealDigits[N[3/4+Log[4],90]]]
```

Equals 2 + Integral_{x=0..1} (1 + x)*log(1 + x) - x dx. - Kritsada Moomuang, May 23 2025
Equals 2 + Sum_{k>=2} (-1)^k/(k*(k+1)*(k-1)). - Davide Rotondo, May 24 2025
Equals Integral_{x=1..2} 2/x + x/2 dx = 3/2 + Integral_{x=1..2} 2/x - x/2 dx. - Jason Bard, Jun 29 2025

A386733 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} {1/(x+y)} dx dy, where {} denotes fractional part.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 01 2025

			0.56382732769577740059825665959334054154152531811711...

Ovidui Furdui, Problem 150, Problems, Missouri J. Math. Sci., Vol. 16, No. 2 (2004), p. 130; Huizeng Qin, Solution to Problem 150, ibid., Vol. 17, No. 3 (2005), pp. 197-199.
Ovidiu Furdui, Multiple Fractional Part Integrals and Euler's Constant, Miskolc Mathematical Notes, Vol. 17, No. 1 (2016), pp. 255-266.

Cf. A016627, A072691, A153810, A345208, A386734, A386735.

RealDigits[2*Log[2] - Pi^2/12, 10, 120][[1]]

PARI
```
2*log(2) - zeta(2)/2
```

Equals 2*log(2) - Pi^2/12 = A016627 - A072691.

A083680 Decimal expansion of (3/2)*log(3/2).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Jun 15 2003

More generally for x > 1: Sum_{k>=1} H(k)/x^k = x/(1-x)*log(1-1/x).

			0.60819766216224657296701967319652370485798563519374...

Index entries for transcendental numbers.

Cf. A001008, A002805, A016627.

RealDigits[(3/2)*Log[3/2], 10, 120][[1]] (* Amiram Eldar, Apr 28 2025 *)

log(3/2)*3/2 \\ Charles R Greathouse IV, May 15 2019

Equals Sum_{k>=1} H(k)/3^k where H(k) is the k-th harmonic number.

A133362 Decimal expansion of 1/(2 log 2).

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Oct 26 2007

PrimePi(n) = A000720(n) => (log n)/(2 log 2) for all n > 2. An elegant proof is given in Kontoyiannis.

Base 4 logarithm of the natural logarithm base. - Alonso del Arte, Aug 31 2014

			0.7213475204444817036799623405009460...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.7 Lengyel's constant p. 319 and Section 5.11 Feller's coin tossing p. 341.

Ioannis Kontoyiannis, Some information-theoretic computations related to the distribution of prime numbers, arXiv:0710.4076 [cs.IT], 2007.
Index entries for transcendental numbers

Cf. A000720, A016627 (reciprocal).

Digits:=100: evalf(0.5/log(2)); # R. J. Mathar, Nov 09 2007

RealDigits[1/(2Log[2]), 10, 128][[1]] (* Alonso del Arte, Aug 31 2014 *)

1/log(4) \\ Charles R Greathouse IV, Mar 24 2016

More terms from R. J. Mathar, Nov 09 2007

A145425 Decimal expansion of Sum_{k>=1} 1/(k*(36k^2-1)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 08 2009

			0.03421279412205515...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, 1996, 4.1.27.

I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products (6th ed.), 2000, (eq. 0.236.3).

Cf. A016627, A156057.

Maple
```
2*ln(2)-3+3/2*ln(3) ;
```

RealDigits[3*Log[3]/2 + 2*Log[2] - 3, 10, 120, -1][[1]] (* Amiram Eldar, Aug 23 2024 *)

Equals A016627+3*(A156057-1).

Equals (3/2) * log(3) + 2*log(2) - 3. - Sean A. Irvine, Apr 05 2025

A157700 Decimal expansion of log(4/(1 + sqrt(2))).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 04 2009

Equals Sum_{n>=2, n even} binomial(2n,n)/(n*4^n) = A016627-A091648.

			0.5049207741003475936018...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. H. Lehmer, Interesting series involving the Central Binomial Coefficient, Am. Math. Monthly 92, no 7 (1985) 449-457.
Index entries for transcendental numbers

SetDefaultRealField(RealField(100)); Log(4/(1+Sqrt(2))); // G. C. Greubel, Oct 02 2018

Maple
```
log(4/(1+sqrt(2))) ;
```

RealDigits[Log[4/(1+Sqrt[2])],10,120][[1]] (* Harvey P. Dale, Jun 08 2014 *)

default(realprecision, 100); log(4/(1+sqrt(2))) \\ G. C. Greubel, Oct 02 2018

A216701 Decimal expansion of sqrt(2)/(2*log(2)).

Original entry on oeis.org

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

Offset: 1

John W. Nicholson, Sep 16 2012

			1.020139446596789481748279105532262769155098088682029887792564377467...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A002193 (sqrt(2)), A016627 (2*log(2)).

SetDefaultRealField(RealField(100)); Sqrt(2)/(2*Log(2)); // G. C. Greubel, Oct 05 2018

RealDigits[Sqrt[2]/(2*Log[2]), 10, 105][[1]] (* T. D. Noe, Sep 17 2012 *)

default(realprecision, 100); sqrt(2)/(2*log(2)) \\ G. C. Greubel, Oct 05 2018

cp's OEIS Frontend

A344041 Decimal expansion of Sum_{k>=1} F(k)/(k*2^k), where F(k) is the k-th Fibonacci number (A000045).

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A352485 Decimal expansion of the probability that when a unit interval is broken at two points uniformly and independently chosen at random along its length the lengths of the resulting three intervals are the altitudes of a triangle.

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A358516 Decimal expansion of Sum_{k >= 1} (-1)^(k+1)*1/((k+2)*(k+3)).

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A358646 Decimal expansion of 3/4 + log(4).

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A386733 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} {1/(x+y)} dx dy, where {} denotes fractional part.

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A083680 Decimal expansion of (3/2)*log(3/2).

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A133362 Decimal expansion of 1/(2 log 2).

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A358516 Decimal expansion of Sum_{k >= 1} (-1)^(k+1)1/((k+2)(k+3)).