A123168 Continued fraction for c = sqrt(2)*(exp(sqrt(2))-1)/(exp(sqrt(2))+1).

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

Author

Benoit Cloitre, Oct 02 2006

Keywords

• nonn
• cofr
• easy

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1500
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A000217, A000384, A212343.

Odd bisection of A062828 with 0 prepended.

Cf. A274540, A123251.

Programs

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 *)

PARI

default(realprecision, 1000); contfrac(sqrt(2)*(exp(sqrt(2))-1)/ (exp(sqrt(2))+1)) \\ Michel Marcus, Oct 11 2016

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