A072334 -id:A072334 - cp's OEIS frontend

A116938 Expansion of e^2 in base 2.

1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0

Offset: 3

Jonathan Vos Post, Mar 21 2006

			111.010001000000 (base 2) ~ 7.389056098930650... (base 10) ~ e^2. 100 decimal places precision here.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.
Eli Maor, e: The Story of a Number, Princeton Univ. Press, 1994.

John Cosgrave, links to "New Proofs of the Irrationality of e^2 and e^4."

Cf. A001113 (e), A072334 (e^2), A090142 (e^2-e).
Cf. A090143 (e^3-2e^2+e/2), A089139 (e^4-3e^3+2e^2-e/6), A090143 (e^3-2e^2+e/2).
Cf. A001671 (powers of e rounded up), A107586 (powers of e^(1/e) rounded up).

RealDigits[E^2, 2, 100] (* Stefan Steinerberger, Mar 30 2006 *)

A236289 Decimal expansion of e^2 / (e - 1).

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Feb 04 2014

			4.3002585353283716597452894764616...

Cf. A001113 (decimal expansion of e), A072334 (decimal expansion of e^2), A091131 (decimal expansion of e-1).

Mathematica
```
RealDigits[E^2/(E-1), 10, 130][[1]]
```

exp(2)/(exp(1)-1) \\ Charles R Greathouse IV, Apr 21 2016

Equals A072334 / A091131.

Mathematica program modified by Harvey P. Dale, Aug 05 2014

A303617 Decimal expansion of Sum_{k >= 0} 2^(2k+1)/Product_{i = 0..k} (2i+1).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Apr 27 2018

			8.83943924091904909456698024436203574171002846378309279604186339401138107...
2/1 + 2^3/(1*3) + 2^5/(1*3*5) + 2^7/(1*3*5*7) + 2^9/(1*3*5*7*9) + 2^11/(1*3*5*7*9*11) + 2^13/(1*3*5*7*9*11*13) + ...

Cf. A001147, A004171.

Cf. A069998, A072334, A110894.

RealDigits[E^2 Sqrt[Pi/2] Erf[Sqrt[2]], 10, 100][[1]]

suminf(k=0, 2^(2*k+1)/prod(i=0, k, (2*i+1))) \\ Michel Marcus, Apr 27 2018

Equals e^2*sqrt(Pi/2)*erf(sqrt(2)) = A072334*A069998*A110894.

A330479 Decimal expansion of 2e^2-2 (or 2(e^2-1)).

Original entry on oeis.org

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

Offset: 2

Eder Vanzei, Dec 15 2019

Solution of the equation W(x+W(x+W(...))) = 2, where W(x) is the Lambert W function.

2*e^2-2 is a transcendental number.

			12.7781121978613004544608549211500156263606311411036946481742556450...

Index entries for transcendental numbers

Cf. A001113, A072334.

RealDigits[2*(E^2-1), 10, 105][[1]] (* Amiram Eldar, Dec 19 2019 *)

Equals 2*(A072334 - 1).

A344144 Decimal expansion of Sum_{n>=0} A000108(n)/n!.

Original entry on oeis.org

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

Offset: 1

Ya-Ping Lu, May 10 2021

			5.09067872931716562297867771990749843797385168020937561652512724311065360258491...

Cf. A000108, A072334, A229020.

RealDigits[E^2 * BesselI[2, 2], 10, 100][[1]] (* Amiram Eldar, May 17 2021 *)

suminf(n=0, (2*n)!/(n!*n!*(n+1)!)) \\ Michel Marcus, May 16 2021

Equals e^2 * BesselI(2,2) = A072334 * A229020. - Amiram Eldar, May 17 2021

A377635 Decimal expansion of 1/(exp(2) - 1).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Nov 05 2024

			0.1565176427496656518180806234654239164560069706202...

Michael Penn, One of my favorite identities, YouTube video, 2023.
Michael Ian Shamos, Shamos's catalog of the real numbers, 2011, p. 218.
Index entries for transcendental numbers
Index entries for sequences related to zeta function.

Cf. A000796, A001113, A072334.

Cf. A027641, A027642, A073747.

First[RealDigits[1/(Exp[2] - 1), 10, 100]]

1/(exp(2) - 1) \\ Amiram Eldar, Nov 08 2024

Equals 1/(A072334 - 1).
Equals Sum_{k >= 1} (-1)^(k+1)*zeta(2*k)/Pi^(2*k).
From Amiram Eldar, Nov 08 2024: (Start)
Formulas from Shamos (2011):
Equals (coth(1) - 1)/2 = (A073747 - 1)/2.
Equals Sum_{k>=1} exp(-2*k).
Equals Sum_{k>=1} 1/(k^2*Pi^2 + 1).
Equals Sum_{k>=0} B(k)*2^(k-1)/k!, where B(k) = A027641(k)/A027642(k) is the k-th Bernoulli number. (End)

cp's OEIS Frontend

A116938 Expansion of e^2 in base 2.

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A236289 Decimal expansion of e^2 / (e - 1).

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A303617 Decimal expansion of Sum_{k >= 0} 2^(2*k+1)/Product_{i = 0..k} (2*i+1).

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A330479 Decimal expansion of 2*e^2-2 (or 2*(e^2-1)).

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A344144 Decimal expansion of Sum_{n>=0} A000108(n)/n!.

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Mathematica

PARI

Formula

A377635 Decimal expansion of 1/(exp(2) - 1).

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Mathematica

PARI

Formula

A303617 Decimal expansion of Sum_{k >= 0} 2^(2k+1)/Product_{i = 0..k} (2i+1).

A330479 Decimal expansion of 2e^2-2 (or 2(e^2-1)).