

A110185


Coefficients of x in the partial quotients of the continued fraction expansion exp(1/x) = [1, x  1/2, 12*x, 5*x, 28*x, 9*x, 44*x, 13*x, ...]. The partial quotients all have the form a(n)*x except the constant term of 1 and the initial partial quotient which equals (x  1/2).


3



0, 1, 12, 5, 28, 9, 44, 13, 60, 17, 76, 21, 92, 25, 108, 29, 124, 33, 140, 37, 156, 41, 172, 45, 188, 49, 204, 53, 220, 57, 236, 61, 252, 65, 268, 69, 284, 73, 300, 77, 316, 81, 332, 85, 348, 89, 364, 93, 380, 97, 396, 101, 412, 105, 428, 109, 444, 113, 460, 117, 476
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OFFSET

0,3


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 1724. [Annotated copy of offprint]


FORMULA

G.f.: x*((1+3*x^2) + 4*x*(3+x^2))/(1x^2)^2 = sum_{n>=0} a(n)*x^n.
Contribution from Carl R. White, Feb 11 2010: (Start)
a(n) = sgn(n) * (2n+1) * (3cos(pi*n)+5)/2.
a(2n+1) = a(2n1) + 4, a(2n+2) = a(2n) + 16, with a(0)=0, a(1)=1, a(2)=12. (End)
a(n) = (5+3(1)^n)(2n1)/2, with a(0)=0. Sum(a(i), i=0..n) = A085787(A042948(n)).  Bruno Berselli, Jan 20 2012


PROG

(PARI) a(n)=polcoeff(x*(1+12*x+3*x^2+4*x^3)/(1x^2)^2+x*O(x^n), n)


CROSSREFS

Cf. A004273, continued fraction expansion of tanh(1), and A204877, continued fraction expansion of 3*tanh(1/3).  Bruno Berselli, Jan 26 2012
Sequence in context: A282578 A205141 A122561 * A299516 A278423 A278661
Adjacent sequences: A110182 A110183 A110184 * A110186 A110187 A110188


KEYWORD

nonn,cofr,easy


AUTHOR

Paul D. Hanna, Jul 14 2005


STATUS

approved



