login
A123168
Continued fraction for c = sqrt(2)*(exp(sqrt(2))-1)/(exp(sqrt(2))+1).
5
0, 1, 6, 5, 14, 9, 22, 13, 30, 17, 38, 21, 46, 25, 54, 29, 62, 33, 70, 37, 78, 41, 86, 45, 94, 49, 102, 53, 110, 57, 118, 61, 126, 65, 134, 69, 142, 73, 150, 77, 158, 81, 166, 85, 174, 89, 182, 93, 190, 97, 198, 101, 206, 105, 214, 109, 222, 113
OFFSET
1,3
COMMENTS
This continued fraction shows exp(sqrt(2)) is irrational (see A274540).
REFERENCES
J. Borwein and D. Bailey, Mathematics by experiment, plausible reasoning in the 21st Century, A. K. Peters, p. 77.
FORMULA
a(2*n) = 4*n-3, a(2*n+1) = 8*n-2.
From Colin Barker, Apr 15 2012: (Start)
a(n) = 2*a(n-2) - a(n-4) for n>5.
G.f.: x^2*(1+6*x+3*x^2+2*x^3)/((1-x)^2*(1+x)^2). (End)
a(n) = (2*n-3)*(3-(-1)^n)/2 for n>1, with a(1) = 0. - Wesley Ivan Hurt, Apr 01 2022
MATHEMATICA
$MinPrecision = 5 $MachinePrecision; ContinuedFraction[Sqrt[2]* (Exp[Sqrt[2]] - 1)/(Exp[Sqrt[2]] + 1), 100] (* G. C. Greubel, Aug 17 2018 *) (* or *)
LinearRecurrence[{0, 2, 0, -1}, {0, 1, 6, 5, 14}, 100] (* Georg Fischer, Aug 26 2022 *)
PROG
(PARI) default(realprecision, 1000); contfrac(sqrt(2)*(exp(sqrt(2))-1)/ (exp(sqrt(2))+1)) \\ Michel Marcus, Oct 11 2016
CROSSREFS
Odd bisection of A062828 with 0 prepended.
Sequence in context: A309550 A274931 A120114 * A119636 A300750 A101493
KEYWORD
nonn,cofr,easy
AUTHOR
Benoit Cloitre, Oct 02 2006
STATUS
approved