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

0, 1, 27, 5, 63, 9, 99, 13, 135, 17, 171, 21, 207, 25, 243, 29, 279, 33, 315, 37, 351, 41, 387, 45, 423, 49, 459, 53, 495, 57, 531, 61, 567, 65, 603, 69, 639, 73, 675, 77, 711, 81, 747, 85, 783, 89, 819, 93, 855, 97, 891, 101, 927, 105, 963, 109, 999, 113

Offset: 0

Author

Bruno Berselli, Jan 23 2012

Keywords

• nonn
• cofr
• easy

Comments

The continued fraction expansions of tanh(1) and 2*tanh(1/2) are in A004273 and A110185, respectively.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• G. Xiao, Contfrac.
• Index entries for continued fractions for constants.
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A004273, A110185.

Programs

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]];

Mathematica

ContinuedFraction[3 Tanh[1/3], 158]
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 *)

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);

PARI

\p232;
       contfrac(3*tanh(1/3))

Formula

G.f.: x*(1+27*x+3*x^2+9*x^3)/((1-x)^2*(1+x)^2).

E.g.f.: 9-4*exp(-x)*(1+2*x)+5*exp(x)*(-1+2*x).

a(n) = (5+4*(-1)^n)*(2*n-1), with a(0)=0.

a(n) = 2*a(n-2)-a(n-4) for n>4.

a(n) = a(n-2)+A040314(n-2) for n>2.

a(n)*a(n+1) = a(2*n^2).

Sum(a(i), i=0..n) = A195162(A042948(n)).