A337668 -id:A337668 - cp's OEIS frontend

A337669 Decimal expansion of Product_{n>=3} (1-1/Fibonacci(n)).

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

Offset: 0

Michel Marcus, Sep 15 2020

			0.18978914361786603634948251429...

Daniel Duverney and Yohei Tachiya, Algebraic independence of certain infinite products involving the Fibonacci numbers, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 97, No. 5 (2021), pp. 29-31; arXiv preprint, arXiv:2009.06250 [math.NT], 2020.
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.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Wikipedia, Dedekind eta function.
Wikipedia, Theta function.

Cf. A000045, A001622, A094214, A337668.

With[{b = 1/GoldenRatio}, RealDigits[(Sqrt[5]/6)*b^(-5/4) * EllipticTheta[2, 0, b] * EllipticTheta[3, 0, b] * EllipticTheta[4, 0, b]/EllipticTheta[4, 0, b^4], 10, 100][[1]]] (* Amiram Eldar, May 27 2021 *)

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

Equals (sqrt(5)/6) * b^(-5/4) * theta_2(b) * theta_3(b) * theta_4(b)/theta_4(b^4), where theta_i are the Jacobi theta functions and b = 1/phi = A094214 (Duverney and Tachiya, 2021). - Amiram Eldar, May 27 2021

Equals (sqrt(5) * phi^(5/4) / 3) * 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). - Amiram Eldar, Mar 26 2024

More terms from Jinyuan Wang, Sep 19 2020

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).

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).

A371529 Decimal expansion of Product_{k>=2} (1 + (-1)^k/Lucas(k)).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 26 2024

			1.07248271775513062588537881652660869304392049333099...

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, A371526, A371527, A371528, A371530.

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

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

Equals Product_{k>=2} (1 + (-1)^k/A000032(k)).
Equals A371525 / (2*sqrt(5)).
Equals (phi^(1/4) / sqrt(5)) * 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).

A371530 Decimal expansion of Product_{k>=1} (1 - (-1)^k/Lucas(k)).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 26 2024

			1.50218004433245676912076258176556998802715258088888...

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, A371526, A371527, A371528, A371529.

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

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

Equals Product_{k>=2} (1 - (-1)^k/A000032(k)).
Equals (2*sqrt(5)) * A371526.
Equals 2 * phi^(1/4) * 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).

A371648 Decimal expansion of Product_{k>=0} (1 + 1/Fibonacci(5^k)).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 31 2024

This constant is a transcendental number (Nyblom, 2004).

			2.40003198933688770409863382912459044885549783193387...

M. A. Nyblom, On the Construction of a Family of Transcendental Valued Infinite Products, Fibonacci Quarterly, Vol. 42, No. 4 (2004), pp. 353-358.
Index entries for transcendental numbers.

Cf. A000045, A145232, A371647.

Similar constants: A337668, A337669, A371650.

RealDigits[Product[1 + 1/Fibonacci[5^k], {k, 0, 10}], 10, 120][[1]]

PARI
```
prodinf(k = 0, 1 + 1/fibonacci(5^k))
```

Equals Product_{k>=0} (1 + 1/A145232(k)).

cp's OEIS Frontend

A337669 Decimal expansion of Product_{n>=3} (1-1/Fibonacci(n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A371529 Decimal expansion of Product_{k>=2} (1 + (-1)^k/Lucas(k)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A371530 Decimal expansion of Product_{k>=1} (1 - (-1)^k/Lucas(k)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI