A123168 Continued fraction for 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, 230, 117, 238

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.

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).

Formula

a(2*n) = 4*n-3, a(2*n+1) = 8*n-2, for n>0.

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

E.g.f.: 3 + 2*x + (4*x - 3)*cosh(x) + 2*(x - 3)*sinh(x). - Stefano Spezia, Oct 23 2025

Mathematica

$MinPrecision = 5 $MachinePrecision; ContinuedFraction[Sqrt[2]* (Exp[Sqrt[2]] - 1)/(Exp[Sqrt[2]] + 1), 100]  (* G. C. Greubel, Aug 17 2018 *)
(* Alternative: *)
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
(PARI) a(n) = if (n%2, if (n==1, 0, 4*n-6), 2*n-3); \\ Bruce Nye, Jun 08 2026

Crossrefs

Cf. A000217, A000384, A212343.

Odd bisection of A062828 with 0 prepended.

Cf. A274540, A123251.

Partial sums: A104584.

Sequence in context: A309550 A274931 A120114 * A119636 A300750 A101493

Adjacent sequences: A123165 A123166 A123167 * A123169 A123170 A123171

Keyword

nonn,cofr,easy,changed

Author

Benoit Cloitre, Oct 02 2006

Status

approved