A019774 -id:A019774 - cp's OEIS frontend

A019798 Decimal expansion of sqrt(2*e).

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

Offset: 1

N. J. A. Sloane

The coefficient a for which y=a*sqrt(x) kisses the exponential function y=exp(x). The kissing point is (0.5, sqrt(e)). For more details, see A257776. Also, inverse of this constant equals the maximum value of sqrt(x)*exp(-x) for positive x, attained at x=1/2. - Stanislav Sykora, Nov 04 2015

			2.3316439815971242033635360621684008763802362991875884230...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers.

Cf. A001113, A019774, A257775, A257776.

SetDefaultRealField(RealField(100)); R:= RealField(); Sqrt(2*Exp(1)); // G. C. Greubel, Sep 08 2018

RealDigits[Sqrt[2*E], 10, 100][[1]] (* G. C. Greubel, Sep 08 2018 *)

sqrt(2*exp(1)) \\ Michel Marcus, Nov 05 2015

From Amiram Eldar, Jul 08 2023: (Start)
Equals Product_{n>=0} (e / (1 + 1/(n-1/2))^n).
Equals Product_{n>=0} (e * (1 - 1/(n+1/2))^n). (End)

A092042 Decimal expansion of e^(1/4).

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Mar 27 2004

e^(1/4) is also the integral from 0 to infinity of e^(-x) * I_0(sqrt(x)), where I_0(z) is a modified Bessel function. - Jean-François Alcover, Mar 10 2011

e^(1/4) maximizes the value of x^(c/(x^4)) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. - A.H.M. Smeets, Aug 16 2018

			1.28402541668774148407342056806243645833....

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
D. M. Bătinetu-Giurgiu, Problem 4133, Crux Mathematicorum, Vol. 42, No. 4 (2016), p. 174; Solution to Problem 4133, ibid., Vol. 43, No. 4 (2017), pp. 167-169.
Index entries for transcendental numbers

Cf. A001113, A019774, A057863, A078688, A092426.

evalf(exp(1/4)); # Muniru A Asiru, Aug 16 2018

RealDigits[(E)^(1/4), 10, 100][[1]] (* Vincenzo Librandi, Mar 01 2013 *)

PARI
```
exp(1/4) \\ Michel Marcus, Jan 19 2017
```

e^(1/4) = 1/2*( 1 +(5 +(9 +(13 +...)/12)/8)/4 ) = 1 +(1 +(1 +(1 +...)/12)/8)/4. - Rok Cestnik, Jan 19 2017
Equals lim_{n->oo} ((2*n-1)!!)^(1/(2*n))/A057863(n)^(1/n^2) (Bătinetu-Giurgiu, 2016). - Amiram Eldar, Apr 10 2022
Equals (Integral_{x=1..oo} 1/(x*log(x)^log(log(x))) dx)/sqrt(Pi). - Kritsada Moomuang, Jun 03 2025

A113011 Decimal expansion of 1/(sqrt(e) - 1).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, following a suggestion of Grover W. Hughes, Oct 09 2005

Has continued fraction 1+2/(3+4/(5+6/7+...)).

Simple continued fraction is 1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, {1, 4k+1, 1}, ..., . - Robert G. Wilson v, Jul 01 2007

			1.54149408253679828413110344447251463834045923684188210947413695663...

G. C. Greubel, Table of n, a(n) for n = 1..20000
Itay Beit-Halachmi and Ido Kaminer, The Ramanujan Library -- Automated Discovery on the Hypergraph of Integer Relations, arXiv:2412.12361 [cs.AI], 2024. See p. 6.
Leonhard Euler, On the formation of continued fractions, arXiv:math/0508227 [math.HO], 2005, see p. 14.
Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18 - 29 mai 2015: Oujda (Maroc).
Eric Weisstein's World of Mathematics, Continued Fraction.
Index entries for transcendental numbers.

Cf. A019774, A113012, A113013.

1/(Sqrt(Exp(1)) - 1); // G. C. Greubel, Apr 09 2018

First@ RealDigits[ 1 / (Exp[1/2] - 1), 10, 111] (* Robert G. Wilson v, Jul 01 2007 *)
f[n_] := Fold[ Last@ #2 + First@ #2/#1 &, 2n - 1, Partition[ Reverse@ Range[ 2n - 2], 2]]; RealDigits[ f[61], 10, 105][[1]] (* Robert G. Wilson v, Jul 07 2012 *)
Rest[realDigitsRecip[Sqrt[E]-1]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Nov 04 2024 *)

1/(sqrt(exp(1)) - 1) \\ G. C. Greubel, Apr 09 2018

Equals Integral_{x = 0..oo} floor(2*x)*exp(-x) dx. - Peter Bala, Oct 09 2013
Equals 3/2 + Sum_{k>=0} tanh(1/2^(k+3))/2^(k+2). - Antonio Graciá Llorente, Jan 21 2024
Conjecture: 1/(sqrt(e) - 1) = 1 + K_{n>=1} 2*n/(4*n^2-1)/1, where K is the Gauss notation for an infinite continued fraction. In the expanded form, 1/(sqrt(e) - 1) = 1 + 2/3/(1 + 4/15/(1 + 6/35/(1 + ...))) (see Beit-Halachmi and Kaminer). - Stefano Spezia, Dec 27 2024
Equals 1/(A019774 - 1). - Hugo Pfoertner, Dec 27 2024

Simpler definition from T. D. Noe, Oct 09 2005

Euler reference from David L. Harden, Oct 09 2005

A334381 Decimal expansion of Sum_{k>=0} 1/(2^k*(k!)^2).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 25 2020

			1/(2^0*0!^2) + 1/(2^1*1!^2) + 1/(2^2*2!^2) + 1/(2^3*3!^2) + ... = 1.56608292975635053729238691...

Index entries for sequences related to Bessel functions or polynomials

Cf. A019774, A055546.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), this sequence (I(0,sqrt(2))), A070910 (I(0,2)).

RealDigits[BesselI[0, Sqrt[2]], 10, 110] [[1]]

suminf(k=0, 1/(2^k*(k!)^2)) \\ Michel Marcus, Apr 26 2020

besseli(0, sqrt(2)) \\ Michel Marcus, Apr 26 2020

Equals BesselI(0,sqrt(2)).

Equals BesselJ(0,sqrt(2)*i). - Jianing Song, Sep 18 2021

A005181 a(n) = ceiling(exp((n-1)/2)).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 91, 149, 245, 404, 666, 1097, 1809, 2981, 4915, 8104, 13360, 22027, 36316, 59875, 98716, 162755, 268338, 442414, 729417, 1202605, 1982760, 3269018, 5389699, 8886111, 14650720, 24154953, 39824785, 65659970, 108254988, 178482301

Offset: 0

N. J. A. Sloane, R. K. Guy

nonn

This sequence illustrates the second law of small numbers because it is a coincidence that its first ten terms are the same as the first ten Fibonacci numbers (A000045). - Alonso del Arte, Mar 18 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, pp. 27 Belin-Pour La Science, Paris 2000.

Vladimir Pletser, Table of n, a(n) for n = 0..1000
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
I. Stewart, Fibonacci Forgeries
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Strong Law of Small Numbers.

Cf. A000045, A019774.

seq(round(ceil(exp((n-1)/2))), n=0..50); # Vladimir Pletser, Sep 15 2013

Table[Ceiling[E^((n - 1)/2)], {n, 0, 39}] (* Alonso del Arte, Mar 18 2013 *)

import math
for n in range(99):
    print(str(int(math.ceil(math.e**((n-1)*0.5)))), end=', ')
# Alex Ratushnyak, Mar 18 2013

Limit_{n->oo} a(n+1)/a(n) = sqrt(e) = 1.64872127... = A019774. - Alois P. Heinz, Feb 19 2019

A few more terms from Alonso del Arte, Mar 18 2013

A249455 Decimal expansion of 2/sqrt(e), a constant appearing in the expression of the asymptotic expected volume V(d) of the convex hull of randomly selected n(d) vertices (with replacement) of a d-dimensional unit cube.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 29 2014

			1.21306131942526684720759906998236090688383627...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 634.

Steven R. Finch, Convex Lattice Polygons, Dec 18 2003. [Cached copy, with permission of the author]
Matthew Perkins and Robert A. Van Gorder, Closed-form calculation of infinite products of Glaisher-type related to Dirichlet series, The Ramanujan Journal, Vol. 49 (2019), pp. 371-389; alternative link. See Corollary 4.3, p. 386.
Index entries for transcendental numbers.

Cf. A019774, A092605, A249456.

Cf. A074962, A243262, A243263.

Mathematica
```
RealDigits[2/Sqrt[E], 10, 100] // First
```

2/exp(.5) \\ Charles R Greathouse IV, Oct 02 2022

Lim_{d -> infinity} V(d) =
  0 if n(d) <= (2/sqrt(e) - epsilon)^d
  1 if n(d) >= (2/sqrt(e) + epsilon)^d.
Equals Product_{m>=1} A(2*m)^((-1)^(m+1)*Pi^(2*m)/(2*m)!), where A(k) is the k-th generalized Glaisher-Kinkelin (or Bendersky-Adamchik) constant (A074962, A243262, A243263, ...) (Perkins and Van Gorder, 2019). - Amiram Eldar, Feb 08 2024

A019775 Decimal expansion of sqrt(e)/2.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.82436063535006407342432539390708178582688805035507...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Pratchayaporn Doemlim, Vichian Laohakoso and Janyarak Tongsomporn, The Continued Fractions of Certain Exponentials, Walailak Journal of Science and Technology, Vol. 16, No. 09, Sept. 2019, pp. 615 - 624.

Cf. A019774.

RealDigits[Sqrt[E]/2,10,120][[1]] (* Harvey P. Dale, Jun 18 2014 *)

From Amiram Eldar, Jul 21 2020: (Start)
Equals Sum_{k>=0} 1/(2^(k+1)*k!).
Equals Sum_{k>=0} 1/(2^k*(k-1)!).
Equals Sum_{k>=0} k/(2*k)!!.
Equals A019774/2. (End)
From Peter Bala, Jun 29 2024: (Start)
Equals Sum_{n >= 0} 1/((1 - 4*n^2)*(2^n)*n!).
Continued fraction expansion [0; 1, 4, 1, 2, 3, 1, 4, 3, 1, ..., 2*n, 3, 1, ...]. (End)

A092514 Decimal expansion of e^(1/5).

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Apr 05 2004

e^(1/5) maximizes the value of x^(c/(x^5)) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. - A.H.M. Smeets, Aug 16 2018

			1.22140275816...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A001113, A019774.

Exp(1/5); // Vincenzo Librandi, Aug 17 2018

evalf(exp(1/5)); # Muniru A Asiru, Aug 16 2018

RealDigits[Surd[E,5],10,120][[1]] (* Harvey P. Dale, Aug 12 2016 *)

PARI
```
exp(1/5) \\ Michel Marcus, Aug 16 2018
```

e^(1/5) = 5^(2*5)/21355775*(1 + Sum_{n>=1} (1 + n^7/5 + n/5)/(5^n*n!)). - Alexander R. Povolotsky, Sep 13 2011

e^(1/5) = (1/2)*lim_{n -> oo} 1 + (6 + (11 + (16 + ... + ((5*n+1)/ (5*n))/...)/15)/10)/5 = lim_{n -> oo} 1 + (1 + (1 + (1 + ... + (1 + 1/(5*n+5))/(5*n)/...)/15)/10)/5. - Rok Cestnik, Jan 19 2017

A092516 Decimal expansion of e^(1/7).

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Apr 05 2004

			1.1535649948951077

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A001113, A019774.

RealDigits[E^(1/7), 10, 50][[1]] (* G. C. Greubel, Feb 19 2017 *)

PARI
```
exp(1/7) \\ G. C. Greubel, Feb 19 2017
```

e^(1/7) = (282475249/1008106751) * (1+ Sum_{n>=1} ((1+n^9/7+n/7)/(7^n*n!))). - Alexander R. Povolotsky, Sep 13 2011

A092615 Decimal expansion of e^(-1/3).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

			0.71653131057378925042560409692537966745311205982147...

Cf. A001113, A019774, A068985, A092041 (reciprocal).

RealDigits[E^(-1/3),10,120][[1]] (* Harvey P. Dale, Feb 13 2012 *)

Equals lim_{x->0} (tanh(x)/x)^(1/x^2). - Amiram Eldar, Jul 04 2022

cp's OEIS Frontend

A019798 Decimal expansion of sqrt(2*e).

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A092042 Decimal expansion of e^(1/4).

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A113011 Decimal expansion of 1/(sqrt(e) - 1).

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A334381 Decimal expansion of Sum_{k>=0} 1/(2^k*(k!)^2).

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A005181 a(n) = ceiling(exp((n-1)/2)).

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A249455 Decimal expansion of 2/sqrt(e), a constant appearing in the expression of the asymptotic expected volume V(d) of the convex hull of randomly selected n(d) vertices (with replacement) of a d-dimensional unit cube.

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A019775 Decimal expansion of sqrt(e)/2.