A002390 -id:A002390 - cp's OEIS frontend

A139345 Decimal expansion of sine of the golden ratio. That is, the decimal expansion of sin((1+sqrt(5))/2).

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

Offset: 0

Mohammad K. Azarian, Apr 15 2008

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

			0.99888450909488479883326824263012904463865119212705...

Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3 (Fall 1998), p. 176; Solution, ibid., Vol. 12, No. 1 (Winter 2000), pp. 61-62.
Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139350.

RealDigits[Sin[GoldenRatio], 10, 100][[1]] (* Amiram Eldar, Feb 07 2022 *)

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

Equals sin(A001622).

Equals 1/A139350. - Amiram Eldar, Feb 07 2022

Leading zero removed by R. J. Mathar, Feb 05 2009

A371525 Decimal expansion of Product_{k>=1} (1 + 1/Lucas(k)).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 26 2024

Any two of the four constants {A337668, A337669, this, A371526} are algebraically independent over Q, while any three are not (Duverney et al., 2022).

			4.79628852318838546381037014075121584981951630809234...

Daniel Duverney, Carsten Elsner, Masanobu Kaneko, and Yohei Tachiya, A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers, Research in Number Theory, Vol. 8 (2022), Article 31; alternative link.
Eric Weisstein's World of Mathematics, Dedekind Eta Function.
Wikipedia, Dedekind eta function.

Cf. A000032, A001622, A002390, A337668, A337669, A371526, A371527, A371528, A371529, A371530.

With[{eta = DedekindEta, tau0 = Log[GoldenRatio]*I/Pi}, RealDigits[2 * Surd[GoldenRatio, 4] * eta[2*tau0]^3 * eta[3*tau0]/(eta[tau0]^2 * eta[4*tau0]), 10, 120][[1]]]

prodinf(k = 1, 1 + 1/(fibonacci(k-1) + fibonacci(k+1)))

Equals Product_{k>=1} (1 + 1/A000032(k)).
Equals 2 * sqrt(5) * A371529.
Equals 2 * phi^(1/4) * eta(2*tau_0)^3 * eta(3*tau_0) / (eta(tau_0)^2 * eta(4*tau_0)), where phi is the golden ratio (A001622), tau_0 = i*log(phi)/Pi, and i = sqrt(-1) (Duverney et al., 2022).

A371526 Decimal expansion of Product_{k>=2} (1 - 1/Lucas(k)).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 26 2024

Any two of the four constants {A337668, A337669, A371525, this} are algebraically independent over Q, while any three are not (Duverney et al., 2022).

			0.33589766935710210314766572663122658048546104021373...

Daniel Duverney, Carsten Elsner, Masanobu Kaneko, and Yohei Tachiya, A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers, Research in Number Theory, Vol. 8 (2022), Article 31; alternative link.
Eric Weisstein's World of Mathematics, Dedekind Eta Function.
Wikipedia, Dedekind eta function.

Cf. A000032, A001622, A002390, A337668, A337669, A371525, A371527, A371528, A371529, A371530.

With[{eta = DedekindEta, tau0 = Log[GoldenRatio]*I/Pi}, RealDigits[(Surd[GoldenRatio, 4] / Sqrt[5]) * eta[2*tau0]^2 * eta[6*tau0]/(eta[3*tau0] * eta[4*tau0]), 10, 120][[1]]]

prodinf(k = 2, 1 - 1/(fibonacci(k-1) + fibonacci(k+1)))

Equals Product_{k>=2} (1 - 1/A000032(k)).
Equals A371530 / (2*sqrt(5)).
Equals (phi^(1/4) / sqrt(5)) * eta(2*tau_0)^2 * eta(6*tau_0) / (eta(3*tau_0) * eta(4*tau_0)), where phi is the golden ratio (A001622), tau_0 = i*log(phi)/Pi, and i = sqrt(-1) (Duverney et al., 2022).

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

A104287 Decimal expansion of log base phi of 2.

Original entry on oeis.org

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

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Feb 28 2005

The fractal dimension of the goldpoint snowflake (Turner, 2003). - Amiram Eldar, Jan 11 2022

			1.4404200904125564790175514995878638024586041426840560816454417295665...

Krassimir Atanassova, Vassia Atanassova, Anthony Shannon and John Turner, New Visual Perspectives on Fibonacci Numbers, World Scientific, 2002, p. 218.

Greg Kuperberg, Breaking the cubic barrier in the Solovay-Kitaev algorithm, QIP2023 video (2023).
J. C. Turner, Some fractals in goldpoint geometry, The Fibonacci Quarterly, Vol. 41, No. 1 (2003), pp. 63-71.
Index entries for transcendental numbers.

Cf. A001622, A002162, A002390, A371176.

RealDigits[Log[2]/Log[GoldenRatio], 10, 100][[1]] (* Amiram Eldar, Nov 24 2020 *)

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

Equals log(2) / log((sqrt(5)+1)/2).

Equals A002162/A002390. - Amiram Eldar, Nov 24 2020

A110609 a(n) = n * binomial(2*n, n-1).

Original entry on oeis.org

0, 1, 8, 45, 224, 1050, 4752, 21021, 91520, 393822, 1679600, 7113106, 29953728, 125550100, 524190240, 2181340125, 9051563520, 37467344310, 154754938800, 637982011590, 2625648168000, 10789623755820, 44277560801760, 181478535620850, 742984788858624, 3038716500907500

Offset: 0

Paul Barry, Jul 30 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.

Cf. A000108, A002390, A002736, A253487.

Column k=1 of A110608.

[0] cat [((4*n+4)*(2*n+1)*Binomial(2*n, n)/(n+2))/2: n in [0..25]]; // Vincenzo Librandi, Jan 09 2015

with(combinat):with(combstruct):a[0]:=0:for n from 1 to 30 do a[n]:=sum((count(Composition(n*2+1),size=n)),j=1..n) od: seq(a[n], n=0..22); # Zerinvary Lajos, May 09 2007
a:=n->sum(sum(binomial(2*n,n)/(n+1), j=1..n),k=1..n): seq(a(n), n=0..22); # Zerinvary Lajos, May 09 2007
series(simplify(x*diff(x*diff((1-sqrt(1-4*x))/(2*x), x), x)), x, 20):
seq(coeff(%, x, k), k=0..18); # Karol A. Penson, Apr 25 2025

Table[CatalanNumber[n]*n^2, {n, 0, 22}] (* Zerinvary Lajos, Jul 08 2009 *)
CoefficientList[Series[x (1 / x^2 - (1 - 6 x + 4 x^2) / ((1 - 4 x)^(3/2) x^2)) / 2, {x, 0, 30}], x] (* Vincenzo Librandi, Jan 09 2015 *)

for(n=0,25, print1(n*binomial(2*n,n-1), ", ")) \\ G. C. Greubel, Sep 01 2017

a(n) = n^2*binomial(2*n, n)/(n+1) = n^2*A000108(n) = A002736(n)/(n+1).
G.f.: -(2*x*(2*x+2*sqrt(1-4*x)-3) - sqrt(1-4*x) + 1)/(2*sqrt((1 - 4*x)^3)*x). - Marco A. Cisneros Guevara, Jul 23 2011; amended by Georg Fischer, Apr 09 2020
(n+1)*(10*n-7)*a(n)+2*n*(5*n-88)*a(n-1) -4*(25*n-22)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 07 2012
From Ilya Gutkovskiy, Jan 20 2017: (Start)
E.g.f.: x*(BesselI(0,2*x) + 2*BesselI(1,2*x) + BesselI(2,2*x))*exp(2*x).
a(n) ~ 4^n*sqrt(n)/sqrt(Pi).
Sum_{n>=1} 1/a(n) = Pi*(2*sqrt(3) + Pi)/18 = 1.152911143694148... (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (2/sqrt(5))*log(phi) + 2*log(phi)^2, where log(phi) = A002390. - Amiram Eldar, Feb 20 2021
G.f.: (x*(d/dx))^2 [g.f. of A000108]. - Karol A. Penson, Apr 25 2025

A139346 Decimal expansion of cosine of the golden ratio, negated. That is, the decimal expansion of -cos((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 15 2008

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

			-0.04722009625435983376687869404879456549554899472734...

Index entries for transcendental numbers.

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

Join[{0}, RealDigits[Cos[GoldenRatio], 10, 100][[1]]] (* Amiram Eldar, Feb 07 2022 *)

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

Equals 1/A139349. - Amiram Eldar, Feb 07 2022

Edited by N. J. A. Sloane, Dec 11 2008

A145433 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*n/binomial(2n,n).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 08 2009

			0.274432715277120323...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.39

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

Cf. A002163, A002390, A145434.

6/25+4/125*5^(1/2)*ln(1/2+1/2*5^(1/2)) ;

RealDigits[6/25 + 4*Sqrt[5]*Log[GoldenRatio]/125, 10, 93] // First (* Jean-François Alcover, Oct 27 2014 *)
RealDigits[Hypergeometric2F1[2, 2, 3/2, -1/4]/2, 10, 93] // First (* Vaclav Kotesovec, Oct 27 2014 *)

Equals 2*(15+2*A002163*A002390)/125.

A371527 Decimal expansion of Product_{k>=2} (1 + (-1)^k/Fibonacci(k)).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 26 2024

			1.13873486170719621809689508574204318763788894791573...

Daniel Duverney, Carsten Elsner, Masanobu Kaneko, and Yohei Tachiya, A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers, Research in Number Theory, Vol. 8 (2022), Article 31; alternative link.
Eric Weisstein's World of Mathematics, Dedekind Eta Function.
Wikipedia, Dedekind eta function.

Cf. A000045, A001622, A002390, A337668, A337669, A371525, A371526, A371528, A371529, A371530.

With[{eta = DedekindEta, tau0 = Log[GoldenRatio]*I/Pi}, RealDigits[2 * Sqrt[5] * Surd[GoldenRatio^5, 4] * eta[tau0]^3 * eta[4*tau0]/eta[2*tau0]^2, 10, 120][[1]]]

PARI
```
prodinf(k = 2, 1 + (-1)^k/fibonacci(k))
```

Equals Product_{k>=2} (1 + (-1)^k/A000045(k)).
Equals 6 * A337669.
Equals 2 * sqrt(5) * phi^(5/4) * eta(tau_0)^3 * eta(4*tau_0) / eta(2*tau_0)^2, where phi is the golden ratio (A001622), tau_0 = i*log(phi)/Pi, and i = sqrt(-1) (Duverney et al., 2022).

A371528 Decimal expansion of Product_{k>=3} (1 - (-1)^k/Fibonacci(k)).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 26 2024

			1.09591388814868203063436944755222157768251662859702...

Daniel Duverney, Carsten Elsner, Masanobu Kaneko, and Yohei Tachiya, A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers, Research in Number Theory, Vol. 8 (2022), Article 31; alternative link.
Eric Weisstein's World of Mathematics, Dedekind Eta Function.
Wikipedia, Dedekind eta function.

Cf. A000045, A001622, A002390, A337668, A337669, A371525, A371526, A371527, A371529, A371530.

With[{eta = DedekindEta, tau0 = Log[GoldenRatio]*I/Pi}, RealDigits[(Surd[GoldenRatio^5, 4] / 3) * eta[4*tau0]/eta[tau0], 10, 120][[1]]]

PARI
```
prodinf(k = 3, 1 - (-1)^k/fibonacci(k))
```

Equals Product_{k>=2} (1 - (-1)^k/A000045(k)).
Equals A337668 / 12.
Equals (phi^(5/4)/3) * eta(4*tau_0) / eta(tau_0), where phi is the golden ratio (A001622), tau_0 = i*log(phi)/Pi, and i = sqrt(-1) (Duverney et al., 2022).

cp's OEIS Frontend

A139345 Decimal expansion of sine of the golden ratio. That is, the decimal expansion of sin((1+sqrt(5))/2).

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A371525 Decimal expansion of Product_{k>=1} (1 + 1/Lucas(k)).

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A371526 Decimal expansion of Product_{k>=2} (1 - 1/Lucas(k)).

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

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A104287 Decimal expansion of log base phi of 2.

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A110609 a(n) = n * binomial(2*n, n-1).

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A139346 Decimal expansion of cosine of the golden ratio, negated. That is, the decimal expansion of -cos((1+sqrt(5))/2).

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