

A274540


Decimal expansion of exp(sqrt(2)).


5



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
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OFFSET

1,1


COMMENTS

Define P(n) = (1/n)*Sum_{k=0..n1} x(nk)*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((C11)/q).
The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.
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



FORMULA

c = exp(sqrt(2)).
c = lim_{n > infinity} P(n) with P(n) = (1/n)*Sum_{k=0..n1} x(nk)*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(nk)*P(k), k=0..n1))) 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



PROG

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


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STATUS

approved



