cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A259147 Decimal expansion of phi(exp(-Pi/2)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 19 2015

Keywords

Examples

			0.74931147780000278742962565878338031190409252790117392831206731...

Links

Crossrefs

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A368211 phi(exp(-Pi/16)), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A259151 phi(exp(-8*Pi)), A363019 phi(exp(-10*Pi)), A363020 phi(exp(-12*Pi)), A292864 phi(exp(-16*Pi)), A363021 phi(exp(-20*Pi)).
Cf. A292819, A292823, A292827.

Programs

  • Mathematica
    phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-Pi/2]], 10, 105] // First

Formula

phi(q) = QPochhammer(q,q) = (q;q)_infinity.
phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
phi(exp(-Pi/2)) = ((sqrt(2) - 1)^(1/3)*(4 + 3*sqrt(2))^(1/24) * exp(Pi/48) * Gamma(1/4))/(2^(5/6)*Pi^(3/4)).
phi(exp(-Pi/2)) = (sqrt(2)-1)^(1/4) * exp(Pi/48) * Gamma(1/4)/(2^(13/16)*Pi^(3/4)). - Vaclav Kotesovec, Jul 03 2017

A292819 Decimal expansion of Product_{k>=1} (1 + exp(-Pi*k/2)).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Sep 24 2017

Keywords

Examples

			1.274394985635799307203533971922303756193562507866370867427400202064220...

Crossrefs

Cf. A259147, A292823, A292827.
Cf. A292820, A292821, A292822.

Programs

  • Mathematica
    RealDigits[E^(Pi/48) / (2^(1/16) * (Sqrt[2]-1)^(1/4)), 10, 120][[1]]
RealDigits[QPochhammer[-1, E^(-Pi/2)]/2, 10, 120][[1]]

Formula

Equals exp(Pi/48) / (2^(1/16) * (sqrt(2)-1)^(1/4)).
Equals A259148 / A259147.

A292823 Decimal expansion of Product_{k>=1} (1 - exp(-Pi*(2*k-1)/2)).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Sep 24 2017

Keywords

Examples

			0.784686075566357581658758601749378057226783036118391503264762538090598...

Crossrefs

Cf. A259147, A292819, A292827.
Cf. A292824, A292825, A292826.

Programs

  • Mathematica
    RealDigits[2^(1/16) * (Sqrt[2]-1)^(1/4) * E^(-Pi/48), 10, 120][[1]]

Formula

Equals 2^(1/16) * (sqrt(2)-1)^(1/4) * exp(-Pi/48).
Equals A259147 / A259148.

A292828 Decimal expansion of Product_{k>=1} (1 + exp(-Pi*(2*k-1))).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Sep 24 2017

Keywords

Examples

			1.043298262644687012527875688815591456103311209998752645741477289470472...

References

  • A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1 (Overseas Publishers Association, Amsterdam, 1986), p. 757, section 6.2.3, formula 5.

Links

Crossrefs

Cf. A259148, A292820, A292824.
Cf. A292827, A292829.

Programs

  • Mathematica
    RealDigits[2^(1/4) * E^(-Pi/24), 10, 120][[1]]

Formula

Equals 2^(1/4) * exp(-Pi/24).
Equals A292820 / A292821.

A292829 Decimal expansion of Product_{k>=1} (1 + exp(-2*Pi*(2*k-1))).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Sep 24 2017

Keywords

Examples

			1.001867449256304435105432750510732150159885395483789405664498761179818...

Crossrefs

Cf. A259149, A292821, A292825.
Cf. A292827, A292828.

Programs

  • Mathematica
    RealDigits[2^(1/16) / (E^(Pi/12) * (Sqrt[2]-1)^(1/4)), 10, 120][[1]]

Formula

Equals 2^(1/16) / (exp(Pi/12) * (sqrt(2)-1)^(1/4)).
Equals A292821 / A292822.
Showing 1-5 of 5 results.

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