A002390 -id:A002390 - cp's OEIS frontend

A139350 Decimal expansion of csc((1+sqrt(5))/2), where (1+sqrt(5))/2 is the golden ratio.

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

Offset: 1

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			1.00111673661465225489616711351705587794461531806624...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139347, A139348, A139349.

RealDigits[Csc[GoldenRatio], 10, 100][[1]] (* Vincenzo Librandi, Feb 19 2013 *)

1/sin((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals 1/A139345. - Amiram Eldar, Feb 07 2022

Edited by Bruno Berselli, Feb 19 2013

A140232 a(n) = ceiling(n*exp((1+sqrt(5))/2)).

Original entry on oeis.org

6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 81, 86, 91, 96, 101, 106, 111, 116, 122, 127, 132, 137, 142, 147, 152, 157, 162, 167, 172, 177, 182, 187, 192, 197, 202, 207, 212, 217, 222, 227, 232, 238, 243, 248, 253, 258, 263, 268, 273, 278, 283

Offset: 1

Mohammad K. Azarian, May 13 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139347, A139348, A139349, A139350, A140231, A016861.

phi:=(1+Sqrt(5))/2; [Ceiling(n*Exp(phi)): n in [1..60]]; // G. C. Greubel, Jun 30 2019

Ceiling[Exp[GoldenRatio]*Range[60]] (* G. C. Greubel, Jun 30 2019 *)

phi=(1+sqrt(5))/2; vector(60, n, ceil(n*exp(phi)) ) \\ G. C. Greubel, Jun 30 2019

[ceil(n*exp(golden_ratio)) for n in (1..60)] # G. C. Greubel, Jun 30 2019

a(n) = ceiling(n*A139341). - R. J. Mathar, Feb 06 2009

A145434 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*n^2/binomial(2n,n).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 08 2009

The numerator in the Apelblat table lacks the square (typo).

			0.125567284722879676...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.40.

Steven Finch, Central Binomial Coefficients [Cached copy, with permission of the author]

Cf. A002163, A002390, A145433.

evalf( 4/25-4/125*5^(1/2)*log(1/2+1/2*5^(1/2)), 120) ;

RealDigits[HypergeometricPFQ[{2, 2, 2}, {1, 3/2}, -1/4]/2, 10, 93] // First
(* or *) RealDigits[4/25 - 4*Sqrt[5]*Log[GoldenRatio]/125, 10, 93] // First (* Jean-François Alcover, Feb 13 2013, updated Oct 27 2014 *)

Equals 4*(5-A002163*A002390)/125.

A152115 Decimal expansion of the dilogarithm of (the golden mean minus 1), Li_2(phi-1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Nov 24 2008

Equals Li_2(phic) = L(phic)-log(phic)*log(1-phic)/2 = A002388/10 - A002390^2, where Li_2(.) is the dilogarithm, L(.) is Roger's dilogarithm, where phic = phi-1 = A094214, where -log(phic)= A002390 = log(1-phic)/2.

			Equals 0.7553956195317414693865200287560823535149635906747...

L. B. W. Jolley, Summation of Series, Dover (1961)

Anatol N. Kirillov, Dilogarithm identities, arXiv:hep-th/9408113.
J. H. Loxton, Special values of the dilogarithm function, Acta Arithm. 43 (2) (1984), 155-166.

RealDigits[ PolyLog[2, (Sqrt[5]-1)/2], 10, 105] // First (* Jean-François Alcover, Feb 12 2013 *)

PARI
```
phic=(sqrt(5)-1)/2 ; dilog(phic);
```

Equals sum_{n>=1} x^n/n^2 for x= 2*sin(Pi/10). [Jolley eq (360d)]

More terms from Jean-François Alcover, Feb 12 2013

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.

Original entry on oeis.org

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).

A145436 Decimal expansion of Sum_{n>=0} (-1)^n/((2n+1)^2*binomial(2n,n)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 08 2009

			0.95023960511664325898162795295142690916973085105890...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.45.
Bruce C. Berndt, Ramanujan's Notebooks, Part 1, Springer-Verlag, 1985, Chapter 9, p. 289, eq. (vii).
Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013. See p. 222.

Cf. A000984, A002390, A013661, A294486.

Maple
```
1/6*Pi^2-3*ln(1/2+1/2*5^(1/2))^2 ;
```

RealDigits[Zeta[2] - 3 * Log[GoldenRatio]^2, 10, 120][[1]] (* Amiram Eldar, May 06 2023 *)

Pi^2/6-3*log(1/2+sqrt(5)/2)^2 \\ Charles R Greathouse IV, Sep 13 2013

Equals A013661 - 3*A002390^2.

A249389 Decimal expansion of the constant 'B' appearing in the asymptotic expression of the number of partitions of n as (B/(2Pin))exp(2B*sqrt(n)), in case of partitions into integers, each of which occurring only an odd number of times.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 27 2014

			1.133841556205496466733768632460501931206029628808654...

G. C. Greubel, Table of n, a(n) for n = 1..10000
F. C. Auluck, K. S. Singwi and B. K. Agarwala, On a new type of partition, Proc. Nat. Inst. Sci. India 16 (1950) 147-156.
Steven Finch, Integer Partitions, September 22, 2004. [Cached copy, with permission of the author]

Cf. A000041, A002390, A055922.

B = Sqrt[Pi^2/12 + 2*Log[GoldenRatio]^2]; RealDigits[B, 10, 99] // First

sqrt(Pi^2/12 + 2*(log((1+sqrt(5))/2))^2) \\ G. C. Greubel, Apr 06 2017

B = sqrt(Pi^2/12 + 2*log(phi)^2), where phi is the golden ratio.

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

Original entry on oeis.org

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).

A384238 Decimal expansion of sqrt(5) - log(phi) - 1, where phi is the golden ratio.

Original entry on oeis.org

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

Offset: 0

Kritsada Moomuang, May 22 2025

			0.75485615244018624891...

Cf. A002163, A002390, A384682

RealDigits[Sqrt[5] - Log[GoldenRatio] - 1, 10, 100, -1][[1]]

Equals Integral_{x=0..1} 2/(1 + sqrt(1 + 4*x)) dx.
Equals Integral_{x=0..1} 1/(1 + x/(1 + x/(1 + x/...))) dx.
Equals A002163 - A002390 - 1.

cp's OEIS Frontend

A139350 Decimal expansion of csc((1+sqrt(5))/2), where (1+sqrt(5))/2 is the golden ratio.

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A140232 a(n) = ceiling(n*exp((1+sqrt(5))/2)).

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A145434 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*n^2/binomial(2n,n).

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A152115 Decimal expansion of the dilogarithm of (the golden mean minus 1), Li_2(phi-1).

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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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A145436 Decimal expansion of Sum_{n>=0} (-1)^n/((2n+1)^2*binomial(2n,n)).

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A249389 Decimal expansion of the constant 'B' appearing in the asymptotic expression of the number of partitions of n as (B/(2*Pi*n))*exp(2*B*sqrt(n)), in case of partitions into integers, each of which occurring only an odd number of times.

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A249389 Decimal expansion of the constant 'B' appearing in the asymptotic expression of the number of partitions of n as (B/(2Pin))exp(2B*sqrt(n)), in case of partitions into integers, each of which occurring only an odd number of times.