A123169 Continued fraction for sqrt(1/2)*(exp(sqrt(1/2))-1)/(exp(sqrt(1/2))+1).

0, 4, 6, 20, 14, 36, 22, 52, 30, 68, 38, 84, 46, 100, 54, 116, 62, 132, 70, 148, 78, 164, 86, 180, 94, 196, 102, 212, 110, 228, 118, 244, 126, 260, 134, 276, 142, 292, 150, 308, 158, 324, 166, 340, 174, 356, 182, 372, 190, 388, 198, 404, 206, 420, 214, 436, 222, 452

Offset: 1

Author

Benoit Cloitre, Oct 02 2006

Keywords

• nonn
• easy

Comments

This continued fraction shows exp(sqrt(1/2)) is irrational.

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..1000
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A123168.

Programs

Magma

[n eq 1 select 0 else (3 + (-1)^n)*(2*n-3): n in [1..100]]; // G. C. Greubel, Jul 19 2023

Maple

A123169:=n->(3+(-1)^n)*(-3+2*n): 0,seq(A123169(n), n=2..100); # Wesley Ivan Hurt, Apr 27 2017

Mathematica

LinearRecurrence[{0,2,0,-1}, {0,4,6,20,14}, 100] (* G. C. Greubel, Jul 19 2023 *)

PARI

a(n)=if(n%2,max(4*n-6,0),8*n-12) \\ Charles R Greathouse IV, Jun 28 2013

SageMath

[(3+(-1)^n)*(2*n-3) + 2*int(n==1) for n in range(1,101)] # G. C. Greubel, Jul 19 2023

Formula

a(1) = 0, for n >= 1, a(2*n) = 16*n - 12, a(2*n+1) = 8*n - 2.

From Colin Barker, Jun 28 2013: (Start)

a(n) = (3 + (-1)^n)*(-3 + 2*n) for n > 1.

a(n) = 2*a(n-2) - a(n-4) for n > 5.

G.f.: 2*x^2*(2+3*x+6*x^2+x^3)/((1-x)^2*(1+x)^2). (End)