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 1-8 of 8 results.

A308716 Decimal expansion of 2*sinh(Pi/2)/Pi.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Jun 19 2019

Keywords

Examples

			1.465052383336634877609179374112651007819934264167...

Crossrefs

Cf. A016742, A060294, A156648, A308715.

Programs

  • Mathematica
    RealDigits[2 Sinh[Pi/2]/Pi, 10, 120][[1]]
  • PARI
    2*sinh(Pi/2)/Pi \\ Michel Marcus, Jun 20 2019

Formula

Equals Product_{k>=1} (1 + 1/(4*k^2)).
Equals Product_{k>=1} (1 + 1/A016742(k)).
Equals binomial(0, i/2), where i is the imaginary unit. - Amiram Eldar, Nov 25 2020

A367960 Decimal expansion of tanh(Pi/2).

Original entry on oeis.org

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

Views

Author

R. J. Mathar, Dec 06 2023

Keywords

Examples

			0.91715233566727434637309...

References

  • Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013. See p. 225.

Crossrefs

Cf. A367961, A367959, A308715, A083124 (cont. frac).

Programs

  • Maple
    evalf(tanh(Pi/2)) ;
  • Mathematica
    First[RealDigits[Tanh[Pi/2],10,100]] (* Paolo Xausa, Dec 06 2023 *)

Formula

Equals 1/A367961 = A367959 / A308715 = (2/Pi)*A228048.
Equals (e^Pi - 1)/(e^Pi + 1) = K_{n>0} Pi^(2-[n=1])/(4*n - 2) (see Clawson at p. 225). - Stefano Spezia, Jul 01 2024

A334402 Decimal expansion of cosh(Pi).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 26 2020

Keywords

Comments

This constant is transcendental.

Examples

			(e^Pi + e^(-Pi))/2 = 11.5919532755215206277517520525601...

Links

Crossrefs

Cf. A000796, A001113, A039661, A068470, A073743, A093580, A175314, A308715, A323982, A334401.

Programs

  • Mathematica
    RealDigits[Cosh[Pi], 10, 110] [[1]]

Formula

Equals Sum_{k>=0} Pi^(2*k)/(2*k)!.
Equals Product_{k>=0} (1 + 4/(2*k+1)^2).
Equals Product_{k>=1} (k^2 + 4)/(k^2 + 1). - Amiram Eldar, Aug 09 2020

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

A367961 Decimal expansion of coth(Pi/2).

Original entry on oeis.org

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

Views

Author

R. J. Mathar, Dec 06 2023

Keywords

Examples

			1.090331410727368230030...

Crossrefs

Cf. A367960, A308715, A367959.

Programs

  • Maple
    evalf(coth(Pi/2)) ;
  • Mathematica
    First[RealDigits[Coth[Pi/2],10,100]] (* Paolo Xausa, Dec 06 2023 *)

Formula

Equals 1/A367960 = A308715 / A367959.

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.

A330865 Decimal expansion of cosh(Pi/2)/Pi.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 28 2020

Keywords

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))/(2*Pi) = 0.7986963159564630848638067...

Crossrefs

Cf. A042972, A049006, A049541, A156648, A175615, A308715, A308716, A308718, A323982, A330864, A334402.

Programs

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

Formula

Equals Sum_{k>=0} Pi^(2*k-1)/(4^k*(2*k)!).
Equals Product_{k>=2} (1 - (-1)^k/k^2).
Equals (i^(-i) + i^i)/(2*Pi), where i is the imaginary unit.

A111195 a(n) = 2^(-n) * Sum_{k=0..n} binomial(2*n+1, 2*k+1) * A000364(k).

Original entry on oeis.org

1, 2, 5, 26, 269, 4666, 121017, 4370722, 209364537, 12833657010, 979336390669, 91018760056938, 10120101446389765, 1326280083965014634, 202311875122389093761, 35535622109342844729074
Offset: 0

Views

Author

Philippe Deléham, Oct 24 2005

Keywords

Crossrefs

Cf. A000364, A103327, A308715.

Programs

  • Mathematica
    t = Range[0, 34]!CoefficientList[ Series[ Sec[x], {x, 0, 34}], x]; f[n_] := 2^(-n)*Sum [Binomial[2n + 1, 2k + 1]*t[[2k + 1]], {k, 0, n}]; Table[ f[n], {n, 0, 17}] (* Robert G. Wilson v, Oct 24 2005 *)
Table[Sum[Binomial[2*n + 1, 2*k + 1]*Abs[EulerE[2*k]], {k, 0, n}] / 2^n, {n, 0, 20}] (* Vaclav Kotesovec, Jul 10 2021 *)

Formula

a(n) ~ cosh(Pi/2) * 2^(3*n + 3) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Jul 10 2021

Extensions

More terms from Robert G. Wilson v, Oct 24 2005
Showing 1-8 of 8 results.

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