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A204877 Continued fraction expansion of 3*tanh(1/3).

%I A204877 #24 Sep 08 2022 08:46:01
%S A204877 0,1,27,5,63,9,99,13,135,17,171,21,207,25,243,29,279,33,315,37,351,41,
%T A204877 387,45,423,49,459,53,495,57,531,61,567,65,603,69,639,73,675,77,711,
%U A204877 81,747,85,783,89,819,93,855,97,891,101,927,105,963,109,999,113
%N A204877 Continued fraction expansion of 3*tanh(1/3).
%C A204877 The continued fraction expansions of tanh(1) and 2*tanh(1/2) are in A004273 and A110185, respectively.
%H A204877 Bruno Berselli, <a href="/A204877/b204877.txt">Table of n, a(n) for n = 0..1000</a>
%H A204877 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>.
%H A204877 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>.
%H A204877 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A204877 G.f.: x*(1+27*x+3*x^2+9*x^3)/((1-x)^2*(1+x)^2).
%F A204877 E.g.f.: 9-4*exp(-x)*(1+2*x)+5*exp(x)*(-1+2*x).
%F A204877 a(n) = (5+4*(-1)^n)*(2*n-1), with a(0)=0.
%F A204877 a(n) = 2*a(n-2)-a(n-4) for n>4.
%F A204877 a(n) = a(n-2)+A040314(n-2) for n>2.
%F A204877 a(n)*a(n+1) = a(2*n^2).
%F A204877 Sum(a(i), i=0..n) = A195162(A042948(n)).
%t A204877 ContinuedFraction[3 Tanh[1/3], 158]
%t A204877 CoefficientList[Series[x (1 + 27 x + 3 x^2 + 9 x^3) / ((1 - x)^2 (1 + x)^2), {x, 0, 60}], x] (* _Vincenzo Librandi_, Jun 14 2013 *)
%o A204877 (PARI) \p232;
%o A204877        contfrac(3*tanh(1/3))
%o A204877 (Magma) I:=[0,1,27,5,63]; [n le 5 select I[n] else 2*Self(n-2)-Self(n-4): n in [1..58]];
%o A204877 (Maxima) makelist(coeff(taylor(x*(1+27*x+3*x^2+9*x^3)/((1-x)^2*(1+x)^2), x, 0, n), x, n), n, 0, 57);
%Y A204877 Cf. A004273, A110185.
%K A204877 nonn,cofr,easy
%O A204877 0,3
%A A204877 _Bruno Berselli_, Jan 23 2012