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.

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A049007 Continued fraction for i^i = exp(-Pi/2).

Original entry on oeis.org

0, 4, 1, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 20, 1, 3, 6, 10, 3, 2, 1, 1, 7, 2, 2, 1, 1, 1, 2, 7, 1, 23, 28, 2, 1, 2, 3, 138, 1, 4, 2, 3, 1, 1, 50, 1, 2, 1, 1, 6, 1, 24, 1, 2, 2, 1, 1, 1, 1, 1, 4, 6, 11, 1, 16, 3, 3, 1, 1, 1, 2, 8, 3, 47, 2, 1, 2, 2, 1, 38, 1, 5, 1, 147
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			0.20787957635076190854695561983497877003387...
i^i = 0.207879576350761908546... = 0 + 1/(4 + 1/(1 + 1/(4 + 1/(3 + ...)))). - _Harry J. Smith_, Apr 28 2009

Links

Crossrefs

Cf. A049006 (decimal expansion).

Programs

  • Mathematica
    ContinuedFraction[ E^(-Pi/2), 100]
  • PARI
    { allocatemem(932245000); default(realprecision, 21000); x=contfrac(exp(-Pi/2)); for (n=1, 20000, write("b049007.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 28 2009

Extensions

Offset changed by Andrew Howroyd, Aug 03 2024

A116186 Decimal expansion of real part of i^(i^i), that is, Re(i^(i^i)).

Original entry on oeis.org

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

Views

Author

Peter C. Heinig (algorithms(AT)gmx.de), Apr 15 2007

Keywords

Comments

If Schanuel's Conjecture is true, then i^i^i is transcendental (see Marques and Sondow 2010, p. 79).

Examples

			i^(i^i) = 0.947158998072378380653475352018 + 0.320764449979308534660116845875 i.

Links

Crossrefs

Cf. A049006, A116191.

Programs

  • Magma
    C := ComplexField(100);  Real(I^I^I); // G. C. Greubel, May 11 2019
  • Mathematica
    RealDigits[ Re[I^I^I], 10, 100] // First
  • PARI
    real(I^I^I) \\ Charles R Greathouse IV, May 15 2013
  • Sage
    numerical_approx((i^i^i).real(), digits=100) # G. C. Greubel, May 11 2019

Formula

Equals cos(Pi/2 * e^(-Pi/2)). - David Ulgenes, Feb 08 2024

A367959 Decimal expansion of sinh(Pi/2).

Original entry on oeis.org

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

Views

Author

R. J. Mathar, Dec 06 2023

Keywords

Examples

			2.301298902307294873463040...

Crossrefs

Cf. A308716, A308715, A049006.

Programs

  • Maple
    evalf(sinh(Pi/2)) ;
  • Mathematica
    First[RealDigits[Sinh[Pi/2],10,100]] (* Paolo Xausa, Dec 06 2023 *)
  • PARI
    sinh(Pi/2) \\ Amiram Eldar, Dec 11 2023

Formula

Equals (Pi/2) * A308716 = A308715 - A049006.
Equals Product_{k>=1} (4*k^2+1)/(4*k^2-1). - Amiram Eldar, Dec 11 2023

A064676 A Beatty sequence for 10*i^i.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 133
Offset: 1

Views

Author

Robert G. Wilson v, Oct 16 2001

Keywords

Comments

ii = 10Chop[ N[ I^I, 24], 10^-30]; Table[ Floor[n*ii], {n, 1, 70} ]

Links

Crossrefs

Cf. A064679, A049006.

Programs

  • PARI
    { default(realprecision, 100); b=10*real(I^I); for (n = 1, 2000, write("b064676.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Sep 21 2009

A116191 Decimal expansion of imaginary part of i^(i^i), that is, Im(i^(i^i)).

Original entry on oeis.org

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

Views

Author

Peter C. Heinig (algorithms(AT)gmx.de), Apr 15 2007

Keywords

Comments

If Schanuel's Conjecture is true, then i^i^i is transcendental (see Marques and Sondow 2010, p. 79).

Examples

			i^(i^i) = 0.947158998072378380653475352018 + 0.320764449979308534660116845875 i.

Links

Crossrefs

Cf. A049006, A116186.

Programs

  • Magma
    C := ComplexField(100);  Im(I^I^I); // G. C. Greubel, May 11 2019
  • Maple
    c := sin((Pi/2)/exp(Pi/2)): Digits := 110: evalf(c, Digits)*10^105:
ListTools:-Reverse(convert(floor(%), base, 10));  # Peter Luschny, Oct 23 2024
  • Mathematica
    RealDigits[ Im[I^I^I], 10, 100] // First
  • PARI
    imag(I^I^I) \\ Charles R Greathouse IV, May 15 2013
  • Sage
    numerical_approx((i^i^i).imag(), digits=100) # G. C. Greubel, May 11 2019

Formula

Equals sin((Pi/2)/exp(Pi/2)). - Peter Luschny, Oct 23 2024

A101748 Decimal expansion of an i^i, namely exp(3*Pi/2).

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Nov 19 2004

Keywords

Comments

This number multiplied by A101749 = A093580.
i^i = exp(-Pi/2 +- 2*k*Pi).

Examples

			This i^i = 111.31777...

Crossrefs

Cf. A049006, A093580, A101749.

Programs

  • Mathematica
    RealDigits[E^(3Pi/2), 10, 111][[1]]

Extensions

Edited by Don Reble, Nov 08 2005

A101749 Decimal expansion of one of the values of i^i, namely exp(-5*Pi/2).

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Nov 19 2004

Keywords

Comments

A101748 multiplied by this number = A093580.
i^i = exp(-Pi/2 +- 2*k*Pi).

Examples

			This i^i = 0.00038820320...

Crossrefs

Cf. A049006, A093580, A101748.

Programs

  • Mathematica
    RealDigits[E^(-5Pi/2), 10, 111][[1]]

Extensions

Edited by Don Reble, Nov 08 2005

A305208 Decimal expansion of the real part of the continued exponential i/Pi.

Original entry on oeis.org

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

Views

Author

Keerthi Vasan Gopala, May 27 2018

Keywords

Comments

This is the real part of e^((i/Pi)*e^((i/Pi)*e^((i/Pi)...))).

Examples

			0.88530292263172060173561116234106499518957753397967...

Crossrefs

Cf. A049006, A077589, A077590, A305200, A305202, A305210.

Programs

  • Mathematica
    Re[Pi*I*N[ProductLog[-I/Pi], 100]]

Formula

Equals Re(Pi*i*LambertW(-i/Pi)).

A305210 Decimal expansion of the imaginary part of continued exponential (i/Pi).

Original entry on oeis.org

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

Views

Author

Keerthi Vasan Gopala, May 27 2018

Keywords

Comments

This is the imaginary part of e^((i/Pi)*e^((i/Pi)*e^((i/Pi)...))).

Examples

			0.256299537163861312529996729880982538078341463884...

Crossrefs

Cf. A049006, A077589, A077590, A305200, A305202, A305208.

Programs

  • Mathematica
    Im[Pi*I*N[ProductLog[-I/Pi], 100]]

Formula

Equals Im(Pi*i*LambertW(-i/Pi)).

A330864 Decimal expansion of sinh(Pi/2)/2.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 28 2020

Keywords

Comments

This constant is transcendental.

Examples

			(1 + 1/2^2) * (1 - 1/3^2) * (1 + 1/4^2) * (1 - 1/5^2) * (1 + 1/6^2) * ... = (e^(Pi/2) - e^(-Pi/2))/4 = 1.15064945115364743673152001...

Links

Crossrefs

Cf. A042972, A049006, A090986, A156648, A175615, A308715, A308716, A308718, A323983, A330865, A334401.

Programs

  • Mathematica
    RealDigits[Sinh[Pi/2]/2, 10, 110] [[1]]
  • PARI
    sinh(Pi/2)/2 \\ Michel Marcus, Apr 28 2020

Formula

Equals Sum_{k>=1} Pi^(2*k-1)/(4^k*(2*k-1)!).
Equals Product_{k>=2} (1 + (-1)^k/k^2).
Equals (i^(-i) - i^i)/4, where i is the imaginary unit.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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