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

1,1

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

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).

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)

Example

c = 2.3227261394604270...

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 *)

Prog

(PARI) {a(n) = (2*n - 1) * 2^(n%2)}; \\ Michael Somos, Feb 04 2012
(Magma) [(2*n-1)*2^(n mod 2): n in [1..50]]; // G. C. Greubel, Jan 27 2018
(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

Crossrefs

Cf. A123168.

Sequence in context: A336091 A347835 A128531 * A333176 A338043 A141670

Adjacent sequences: A123164 A123165 A123166 * A123168 A123169 A123170

Keyword

nonn,cofr

Author

Benoit Cloitre, Oct 02 2006

Status

approved