A086465 -id:A086465 - cp's OEIS frontend

A002390 Decimal expansion of natural logarithm of golden ratio.

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

Offset: 0

N. J. A. Sloane

The Baxa article proves that every gamma >= this constant is the Lévy constant of a transcendental number. - Michel Marcus, Apr 09 2016
The entropy of the golden mean shift. See Capobianco link. - Michel Marcus, Jan 19 2019
Also the limiting value of the area of the function y = 1/x bounded by the abscissa of consecutive F(n) points (where F(n)=A000045(n) are the Fibonacci numbers and n > 0). - Burak Muslu, May 09 2021

			0.481211825059603447497758913424368423135184334385660519661...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 236.
W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 31-38.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..5000
Alexander Adamchuk's comment, Sep 01 2006 Mathematics in Russian
Christoph Baxa, Lévy constants of transcendental numbers, Proc. Amer. Math. Soc. 137 (2009), 2243-2249.
Christopher Brown, The natural logarithm of the golden section, Fibonacci Quarterly 55:5 (2017), pp. 42-44.
Silvio Capobianco, Introduction to Symbolic Dynamics. Part 4: Entropy; The entropy of the golden mean shift, Institute of Cybernetics at TUT; May 12 2010. Slides 15-17.
Simon Plouffe, Plouffe's Inverter, ln(phi) to 10000 digits
Simon Plouffe, ln(0.5+0.5*SQRT(5)) to 2000 digits
Eric Weisstein's World of Mathematics, Fibonacci Hyperbolic Functions
Index entries for transcendental numbers

Cf. A000108, A001622, A013661, A086463, A086466, A263401.

Maple
```
arcsinh(1/2); evalf(%, 120);
```

RealDigits[N[Log[GoldenRatio],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

asinh(1/2) \\ Charles R Greathouse IV, Jan 04 2016

Also equals arcsinh(1/2).
Equals sqrt(5)* A086466 /2. - Seiichi Kirikami, Aug 20 2011
Equals sqrt(5)*(5* A086465 -1)/4. - Jean-François Alcover, Apr 29 2013
Also equals (125*C - 55) / (24*sqrt(5)), where C = Sum_{k>=1} (-1)^(k+1)*1/Cat(k), where Cat(k) = (2k)!/k!/(k+1)! = A000108(k) - k-th Catalan number. See Sep 01 2006 comment at ref. Mathematics in Russian. - Alexander Adamchuk, Dec 27 2013
Equals sqrt(5)/4 * Sum_{n>=0} (-1)^n/((2n+1)*C(2*n,n)) = sqrt(5) *A344041 /4. - Alexander Adamchuk, Dec 27 2013
Equals sqrt((Pi^2/6 - W)/3), where W = Sum_{n>=0} (-1)^n/((2n+1)^2*C(2*n,n)) = A145436, attributed by Alexander Adamchuk to Ramanujan. See Sep 01 2006 comment at ref. Mathematics in Russian. - Alexander Adamchuk, Dec 27 2013
Equals lim_{j->infinity} Sum_{k=F(j)..F(j+1)-1} (1/k), where F = A000045, the Fibonacci sequence. Convergence is slow. For example: Sum_{k=21..33} (1/k) = 0.4910585.... - Richard R. Forberg, Aug 15 2014
Equals Sum_{k>=1} cos(Pi*k/5)/k. - Amiram Eldar, Aug 12 2020
Equals real solution to exp(x)+exp(2*x) = exp(3*x). - Alois P. Heinz, Jul 14 2022
Equals arccoth(sqrt(5)). - Amiram Eldar, Feb 09 2024
Sum_{n >= 1} 1/(n*P(n, sqrt(5))*P(n-1, sqrt(5))), where P(n, x) denotes the n-th Legendre polynomial. The first ten terms of the series gives the approximation log((1 + sqrt(5))/2) = 0.481211825059(39..), correct to 12 decimal places. - Peter Bala, Mar 16 2024
Equals Sum_{n>=0} ((-1)^(n)*binomial(2*n, n))/(2^(4*n + 1)*(2*n + 1)). - Antonio Graciá Llorente, Nov 13 2024

A086466 Decimal expansion of 2*sqrt(5)/5 arccsch(2).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 21 2003

Equals the value of the Dirichlet L-series of the non-principal character modulo 5 (A080891) at s=1. - Jianing Song, Nov 16 2019

			0.43040894096400403888943323295060542542457...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.2, p. 7.

R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, Table 22 for L(m=5,r=3,s=1).
H.-J. Seiffert, Problem B-771, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 32, No. 4 (1994), p. 374; More Sums, Solution to Problem B-771 by Don Redmond, ibid., Vol. 33, No. 5 (1995), pp. 470-471.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Index entries for transcendental numbers

Cf. A086465, A086467, A086468.

Cf. A000032, A000045, A080891, A328717 (s=2), A344041.

2*Log[GoldenRatio]/Sqrt[5] // RealDigits[#, 10, 102]& // First (* Jean-François Alcover, Apr 18 2014 *)

2*log((1+sqrt(5))/2)/sqrt(5) \\ Stefano Spezia, Oct 15 2024

Equals Sum_{k>=1} (-1)^(k-1)/(k*binomial(2*k,k)).
Equals A010532 * A002390 / 10. - R. J. Mathar, Jul 26 2010
Also equals f'(0) = 2*log(phi)/sqrt(5), with f(x) = (phi^x-cos(Pi*x)*phi^-x)/sqrt(5), the real Fibonacci interpolating function. - Jean-François Alcover, Apr 04 2014
Equals Sum_{k>=1} A080891(k)/k = Sum_{k>=1} Kronecker(5,k)/k = 1 - 1/2 - 1/3 + 1/4 + 1/6 - 1/7 - 1/8 + 1/9 + ... - Jianing Song, Nov 16 2019
Equals Sum_{k>=1} F(k)/(k*2^(k+1)), where F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Aug 10 2020
Sum_{k>=1} (2*k+1)*Lucas(k)/(k*(k+1)*2^k) = 10*c + 2 = 6.3040894096... where c is this constant (Seiffert, 1994). - Amiram Eldar, Jan 15 2022
Equals Sum_{k>=1} F(k)/(k*3^k), where F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Jul 02 2023
Equals 1/Product_{p prime} (1 - Kronecker(5,p)/p), where Kronecker(5,p) = 0 if p = 5, 1 if p == 1 or 4 (mod 5) or -1 if p == 2 or 3 (mod 5). - Amiram Eldar, Dec 17 2023
Equals A344041/2. - Hugo Pfoertner, Oct 16 2024

A086467 Decimal expansion of 2*arccsch(2)^2.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 21 2003

			0.4631296...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 77.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27
Eric Weisstein's World of Mathematics, Central Binomial Coefficient
Index entries for transcendental numbers

Cf. A086465, A086466, A086468.

RealDigits[2ArcCsch[2]^2,10,120][[1]] (* Harvey P. Dale, Mar 07 2012 *)

suminf(n=1, (-1)^(n-1)/n^2/binomial(2*n,n)) \\ Michel Marcus, Jul 31 2015

2*asinh(.5)^2 \\ Charles R Greathouse IV, Nov 21 2024

Equals Sum_{n>=1} (-1)^(n-1)/n^2/binomial(2*n,n).
Equals Integral_{x=0..1} log(1+x-x^2)/x dx. - Vaclav Kotesovec, Jun 13 2021
Equals 2*A002390^2. - R. J. Mathar, Jun 07 2024

A086468 Decimal expansion of 2*zeta(3)/5.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 21 2003

			0.48082276126383771415989526460457999630599451693620...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.46.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 446.
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.

Cf. A002117, A086465, A086466, A086467.

SetDefaultRealField(RealField(250)); L:=RiemannZeta();  2*Evaluate(L,3)/5; // G. C. Greubel, Nov 02 2018

First[RealDigits[N[2*Zeta[3]/5, 100]]] (* Stefano Spezia, Nov 02 2018 *)

2*zeta(3)/5 \\ Michel Marcus, Nov 02 2018

Equals Sum_{n>=1} (-1)^(n-1)/(n^3*binomial(2*n,n)).
Equals 2*A002117/5. - R. J. Mathar, Feb 08 2009
Equals (1/10)*Sum_{k>=1} (30*k - 11)/((2*k - 1)*k^3*binomial(2*k,k)^2) (see Finch). - Stefano Spezia, Nov 01 2024

A378807 Decimal expansion of Sum_{k>=1} (-1)^k/binomial(4*k, k) (negated).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Dec 07 2024

			-0.21833954717793443687099832102788539198304864029226...

Necdet Batir and Anthony Sofo, On some series involving reciprocals of binomial coefficients, Appl. Math. Comp., Vol. 220 (2013), pp. 331-338.

Cf. A005810, A086465, A229703, A378806.

RealDigits[HypergeometricPFQ[{1, 4/3, 5/3, 2}, {5/4, 3/2, 7/4}, -27/256]/ 4, 10, 120][[1]]

Equals 4F3(1, 4/3, 5/3, 2; 5/4, 3/2, 7/4; -27/256) / 4, where 4F3 is a generalized hypergeometric function.

Equals 27*d^2/((d^2-4)*(2*d^2+1)^2) + (3*d*(d^2-1)*(2*d^2-1)/(2*(2*d^2+1)^3)) * log(abs((d-1)/(d+1))) + (3*(d^2-1)*(2*d^4-2*d^3-7*d^2-3*d+1)/(4*d*(2*d^2+1)^3)) * (d/(d+2))^(3/2) * arctan(2*sqrt(d^2+2*d)/(d^2+2*d-1)) - (3*(d^2-1)*(2*d^4+2*d^3-7*d^2+3*d+1)/(4*d*(2*d^2+1)^3)) * (d/(d-2))^(3/2) * arctan(2*sqrt(d^2-2*d)/(d^2-2*d-1)), where d = sqrt(1 - (8/sqrt(3))*(((3*sqrt(3)+sqrt(283))/16)^(1/3) - (((3*sqrt(3)+sqrt(283))/16)^(-1/3)))) (Batir and Sofo, 2013, pp. 336-337, Example 4).

A307086 Decimal expansion of 4(5 - sqrt(5)log(phi))/25, where phi is the golden ratio (A001622).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 23 2019

Decimal expansion of the alternating sum of the reciprocals of the central binomial coefficients (A000984).

			1/1 - 1/2 + 1/6 - 1/20 + 1/70 - 1/252 + ... = 0.62783642361439838444422670681975782983017172698388...

Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27
Eric Weisstein's World of Mathematics, Central Binomial Coefficient
Index entries for transcendental numbers

Cf. A000984, A001622, A073016, A086465, A091682, A145433, A145434.

RealDigits[4 (5 - Sqrt[5] Log[GoldenRatio])/25, 10, 120][[1]]

4*(5 - sqrt(5)*log((sqrt(5)+1)/2))/25 \\ Charles R Greathouse IV, May 15 2019

Equals Sum_{k>=0} (-1)^k/binomial(2*k,k).

Equals Sum_{k>=0} (-1)^k*(k!)^2/(2*k)!.

cp's OEIS Frontend

A002390 Decimal expansion of natural logarithm of golden ratio.

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A086466 Decimal expansion of 2*sqrt(5)/5 arccsch(2).

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A086467 Decimal expansion of 2*arccsch(2)^2.

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A086468 Decimal expansion of 2*zeta(3)/5.

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A378807 Decimal expansion of Sum_{k>=1} (-1)^k/binomial(4*k, k) (negated).

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A307086 Decimal expansion of 4*(5 - sqrt(5)*log(phi))/25, where phi is the golden ratio (A001622).

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A307086 Decimal expansion of 4(5 - sqrt(5)log(phi))/25, where phi is the golden ratio (A001622).