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

%I A123169 #22 Jul 20 2023 11:15:53
%S A123169 0,4,6,20,14,36,22,52,30,68,38,84,46,100,54,116,62,132,70,148,78,164,
%T A123169 86,180,94,196,102,212,110,228,118,244,126,260,134,276,142,292,150,
%U A123169 308,158,324,166,340,174,356,182,372,190,388,198,404,206,420,214,436,222,452
%N A123169 Continued fraction for sqrt(1/2)*(exp(sqrt(1/2))-1)/(exp(sqrt(1/2))+1).
%C A123169 This continued fraction shows exp(sqrt(1/2)) is irrational.
%D A123169 J. Borwein and D. Bailey, Mathematics by experiment, plausible reasoning in the 21st Century, A. K. Peters, p. 77
%H A123169 G. C. Greubel, <a href="/A123169/b123169.txt">Table of n, a(n) for n = 1..1000</a>
%H A123169 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A123169 a(1) = 0, for n >= 1, a(2*n) = 16*n - 12, a(2*n+1) = 8*n - 2.
%F A123169 From _Colin Barker_, Jun 28 2013: (Start)
%F A123169 a(n) = (3 + (-1)^n)*(-3 + 2*n) for n > 1.
%F A123169 a(n) = 2*a(n-2) - a(n-4) for n > 5.
%F A123169 G.f.: 2*x^2*(2+3*x+6*x^2+x^3)/((1-x)^2*(1+x)^2). (End)
%p A123169 A123169:=n->(3+(-1)^n)*(-3+2*n): 0,seq(A123169(n), n=2..100); # _Wesley Ivan Hurt_, Apr 27 2017
%t A123169 LinearRecurrence[{0,2,0,-1}, {0,4,6,20,14}, 100] (* _G. C. Greubel_, Jul 19 2023 *)
%o A123169 (PARI) a(n)=if(n%2,max(4*n-6,0),8*n-12) \\ _Charles R Greathouse IV_, Jun 28 2013
%o A123169 (Magma) [n eq 1 select 0 else (3 + (-1)^n)*(2*n-3): n in [1..100]]; // _G. C. Greubel_, Jul 19 2023
%o A123169 (SageMath) [(3+(-1)^n)*(2*n-3) + 2*int(n==1) for n in range(1,101)] # _G. C. Greubel_, Jul 19 2023
%Y A123169 Cf. A123168.
%K A123169 nonn,easy
%O A123169 1,2
%A A123169 _Benoit Cloitre_, Oct 02 2006