A078688 -id:A078688 - cp's OEIS frontend

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

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

A267318 Continued fraction expansion of e^(1/5).

Original entry on oeis.org

1, 4, 1, 1, 14, 1, 1, 24, 1, 1, 34, 1, 1, 44, 1, 1, 54, 1, 1, 64, 1, 1, 74, 1, 1, 84, 1, 1, 94, 1, 1, 104, 1, 1, 114, 1, 1, 124, 1, 1, 134, 1, 1, 144, 1, 1, 154, 1, 1, 164, 1, 1, 174, 1, 1, 184, 1, 1, 194, 1, 1, 204, 1, 1, 214, 1, 1, 224, 1, 1, 234, 1, 1, 244, 1, 1, 254, 1, 1, 264, 1, 1

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

e^(1/5) is a transcendental number.

In general, the ordinary generating function for the continued fraction expansion of e^(1/k), with k = 1, 2, 3..., is (1 + (k - 1)*x + x^2 - (k + 1)*x^3 + 7*x^4 - x^5)/(1 - x^3)^2.

			e^(1/5) = 1 + 1/(4 + 1/(1 + 1/(1 + 1/(14 + 1/(1 + 1/...))))).

Eric Weisstein's World of Mathematics, e Continued Fraction
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A092514.

Cf. continued fraction expansion of e^(1/k): A003417 (k=1), A058281 (k=2), A078689 (k=3), A078688 (k=4), this sequence (k=5).

[1+(3+10*Floor(n/3))*(1-(n-1)^2 mod 3): n in [0..90]]; // Bruno Berselli, Feb 04 2016

ContinuedFraction[Exp[1/5], 82]
LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 4, 1, 1, 14, 1}, 82]
CoefficientList[Series[(1 + 4 x + x^2 - x^3 + 6 x^4 - x^5) / (x^3 - 1)^2, {x, 0, 70}], x] (* Vincenzo Librandi, Jan 13 2016 *)
Table[1 + (3 + 10 Floor[n/3]) (1 - Mod[(n - 1)^2, 3]), {n, 0, 90}] (* Bruno Berselli, Feb 04 2016 *)

G.f.: (1 + 4*x + x^2 - x^3 + 6*x^4 - x^5)/(1 - x^3)^2.

a(n) = 1 + (3 + 10*floor(n/3))*(1 - (n-1)^2 mod 3). [Bruno Berselli, Feb 04 2016]

Edited by Bruno Berselli, Feb 04 2016

cp's OEIS Frontend

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

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