cp's OEIS Frontend

A274540 Decimal expansion of exp(sqrt(2)).

%I A274540 #40 Nov 21 2024 09:28:31
%S A274540 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,
%T A274540 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,
%U A274540 0,9,7,8,5,8,3,0,8,0,7,3,2,7,9,1,6,5,0
%N A274540 Decimal expansion of exp(sqrt(2)).
%C A274540 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).
%C A274540 The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.
%C A274540 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)).
%C A274540 From _Peter Bala_, Oct 23 2019: (Start)
%C A274540 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.
%C A274540 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)
%H A274540 G. C. Greubel, <a href="/A274540/b274540.txt">Table of n, a(n) for n = 1..10000</a>
%H A274540 The Dev Team and Simon Plouffe, <a href="https://wayback.cecm.sfu.ca/projects/ISC/ISCmain.html">The Inverse Symbolic Calculator (ISC)</a>.
%H A274540 Todd Trimble and Vishal Lama, <a href="https://topologicalmusings.wordpress.com/2008/08/04/continued-fraction-for-e/">Continued fraction for e</a>, Todd and Vishal’s blog 2008/08/04
%H A274540 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A274540 c = exp(sqrt(2)).
%F A274540 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.
%e A274540 c = 4.113250378782927517173581815140304502401663943151...
%p A274540 Digits := 80: evalf(exp(sqrt(2))); # End program 1.
%p A274540 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.
%t A274540 First@ RealDigits@ N[Exp[Sqrt@ 2], 80] (* _Michael De Vlieger_, Jun 27 2016 *)
%o A274540 (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
%Y A274540 Cf. A274541, A274542, A036039, A135002, A135004, A234473.
%Y A274540 Cf. A014176, A002193, A010503, A131594, A020765, A010466, A020775.
%K A274540 cons,nonn
%O A274540 1,1
%A A274540 _Johannes W. Meijer_, Jun 27 2016
%E A274540 More terms from _Jon E. Schoenfield_, Mar 15 2018