A123167 Continued fraction for c=sqrt(2)*(exp(sqrt(2))+1)/(exp(sqrt(2))-1). a(2*n-1) = 8*n-6, a(2*n) = 4*n-1.

2, 3, 10, 7, 18, 11, 26, 15, 34, 19, 42, 23, 50, 27, 58, 31, 66, 35, 74, 39, 82, 43, 90, 47, 98, 51, 106, 55, 114, 59, 122, 63, 130, 67, 138, 71, 146, 75, 154, 79, 162, 83, 170, 87, 178, 91, 186, 95, 194, 99, 202, 103, 210, 107, 218, 111, 226, 115, 234, 119, 242, 123

Offset: 1

Author

Benoit Cloitre, Oct 02 2006

Keywords

• nonn
• cofr

Comments

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

If a(0)=-1 and offset 0: a(6*n) - a(6*n+1) + a(6*n+2) = 0, a(6*n +3) - 4*a(6*n+4) + a(6*n+5) = 0.

Conjecture: Numerator of 4/n - 2/n^2. - Wesley Ivan Hurt, Jul 11 2016

Examples

			c = 2.3227261394604270...
		

References

• J. Borwein and D. Bailey, Mathematics by experiment, plausible reasoning in the 21st Century, A. K. Peters, p. 77
• J. Borwein and K. Devlin, The computer as crucible: an introduction to experimental mathematics, A. K. Peters 2009, p. 91.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A123168.

Programs

GAP

a := [2,3,10,7];; for n in [5..10^3] do a[n] := 2*a[n-2] - a[n-4]; od; a; # Muniru A Asiru, Jan 28 2018

Magma

[(2*n-1)*2^(n mod 2): n in [1..50]]; // G. C. Greubel, Jan 27 2018

Maple

A123167 := proc(n)
    if type(n,'even') then
        2*n-1 ;
    else
        4*n-2 ;
    end if;
end proc: # R. J. Mathar, Jul 25 2013

Mathematica

a[ n_] := (2 n - 1) 2^Mod[n, 2]; (* Michael Somos, Apr 25 2015 *)

PARI

{a(n) = (2*n - 1) * 2^(n%2)}; \\ Michael Somos, Feb 04 2012

Formula

a(n) = - A123168(2 - n) for all n in Z unless n = 1. - Michael Somos, Feb 24 2012

From Colin Barker, Feb 08 2012: (Start)

Empirical g.f.: x*(2+3*x+6*x^2+x^3)/(1-2*x^2+x^4).

Empirical a(n) = 2*a(n-2) - a(n-4). (End)