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A274540 Decimal expansion of exp(sqrt(2)). 4
4, 1, 1, 3, 2, 5, 0, 3, 7, 8, 7, 8, 2, 9, 2, 7, 5, 1, 7, 1, 7, 3, 5, 8, 1, 8, 1, 5, 1, 4, 0, 3, 0, 4, 5, 0, 2, 4, 0, 1, 6, 6, 3, 9, 4, 3, 1, 5, 1, 1, 0, 9, 6, 1, 0, 0, 6, 8, 3, 6, 4, 7, 0, 9, 8, 5, 1, 5, 0, 9, 7, 8, 5, 8, 3, 0, 8, 0, 7, 3, 2, 7, 9, 1, 6, 5, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Define P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k) for n >= 1, and P(0) = 1, with x(q) = C1 and x(n) = 1 for all other n. We find that C2 = lim_{n -> infinity} P(n) = exp((C1-1)/q).

The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.

Some transform pairs: C1 = A002162 (log(2)) and C2 = A135002 (2/exp(1)); C1 = A016627 (log(4)) and C2 = A135004 (4/exp(1)); C1 = A001113 (exp(1)) and C2 = A234473 (exp(exp(1)-1)).

From Peter Bala, Oct 23 2019: (Start)

The constant is irrational: Henry Cohn gives the following proof in Todd and Vishals Blog - "By the way, here's my favorite application of the tanh continued fraction: exp(sqrt(2)) is irrational.

Consider sqrt(2)*(exp(sqrt(2))-1)/(exp(sqrt(2))+1). If exp(sqrt(2)) were rational, or even in Q(sqrt(2)), then this expression would be in Q(sqrt(2)). However, it is sqrt(2)*tanh(1/sqrt(2)), and the tanh continued fraction shows that this equals [0,1,6,5,14,9,22,13,...]. If it were in Q(sqrt(2)), it would have a periodic simple continued fraction expansion, but it doesn't." (End)

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

The Dev Team and Simon Plouffe, The Inverse Symbolic Calculator (ISC).

Todd Trimble and Vishal Lama, Continued fraction for e, Todd and Vishal’s blog 2008/08/04

FORMULA

c = exp(sqrt(2)).

c = lim_{n -> infinity} P(n) with P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k) for n >= 1, and P(0) = 1, with x(1) = (1 + sqrt(2)), the silver mean A014176, and x(n) = 1 for all other n.

EXAMPLE

c = 4.113250378782927517173581815140304502401663943151...

MAPLE

Digits := 80: evalf(exp(sqrt(2))); # End program 1.

P := proc(n) : if n=0 then 1 else P(n) := expand((1/n)*(add(x(n-k)*P(k), k=0..n-1))) fi; end: x := proc(n): if n=1 then (1 + sqrt(2)) else 1 fi: end: Digits := 49; evalf(P(120)); # End program 2.

MATHEMATICA

First@ RealDigits@ N[Exp[Sqrt@ 2], 80] (* Michael De Vlieger, Jun 27 2016 *)

PROG

(PARI) my(x=exp(sqrt(2))); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", ")) \\ Felix Fröhlich, Jun 27 2016

CROSSREFS

Cf. A274541, A274542, A036039, A135002, A135004, A234473.

Cf. A014176, A002193, A010503, A131594, A020765, A010466, A020775.

Sequence in context: A016524 A087963 A316586 * A010323 A261790 A174834

Adjacent sequences:  A274537 A274538 A274539 * A274541 A274542 A274543

KEYWORD

cons,nonn

AUTHOR

Johannes W. Meijer, Jun 27 2016

EXTENSIONS

More terms from Jon E. Schoenfield, Mar 15 2018

STATUS

approved

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Last modified November 28 06:42 EST 2021. Contains 349401 sequences. (Running on oeis4.)