A123167 - OEIS

%I #44 Sep 08 2022 08:45:28

%S 2,3,10,7,18,11,26,15,34,19,42,23,50,27,58,31,66,35,74,39,82,43,90,47,

%T 98,51,106,55,114,59,122,63,130,67,138,71,146,75,154,79,162,83,170,87,

%U 178,91,186,95,194,99,202,103,210,107,218,111,226,115,234,119,242,123

%N 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.

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

%C 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.

%C Conjecture: Numerator of 4/n - 2/n^2. - _Wesley Ivan Hurt_, Jul 11 2016

%D J. Borwein and D. Bailey, Mathematics by experiment, plausible reasoning in the 21st Century, A. K. Peters, p. 77

%D J. Borwein and K. Devlin, The computer as crucible: an introduction to experimental mathematics, A. K. Peters 2009, p. 91.

%H G. C. Greubel, <a href="/A123167/b123167.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).

%F a(n) = - A123168(2 - n) for all n in Z unless n = 1. - _Michael Somos_, Feb 24 2012

%F From _Colin Barker_, Feb 08 2012: (Start)

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

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

%e c = 2.3227261394604270...

%p A123167 := proc(n)

%p     if type(n,'even') then

%p         2*n-1 ;

%p     else

%p         4*n-2 ;

%p     end if;

%p end proc: # _R. J. Mathar_, Jul 25 2013

%t a[ n_] := (2 n - 1) 2^Mod[n, 2]; (* _Michael Somos_, Apr 25 2015 *)

%o (PARI) {a(n) = (2*n - 1) * 2^(n%2)}; \\ _Michael Somos_, Feb 04 2012

%o (Magma) [(2*n-1)*2^(n mod 2): n in [1..50]]; // _G. C. Greubel_, Jan 27 2018

%o (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

%Y Cf. A123168.

%K nonn,cofr

%O 1,1

%A _Benoit Cloitre_, Oct 02 2006