A165998 Denominators of Taylor series expansion of 1/(3*x)*log((1+x)/(1-x)^2).

1, 6, 3, 12, 5, 18, 7, 24, 9, 30, 11, 36, 13, 42, 15, 48, 17, 54, 19, 60, 21, 66, 23, 72, 25, 78, 27, 84, 29, 90, 31, 96, 33, 102, 35, 108, 37, 114, 39, 120, 41, 126, 43, 132, 45, 138, 47, 144, 49, 150, 51, 156, 53, 162, 55, 168, 57, 174, 59, 180, 61, 186, 63, 192, 65, 198

Offset: 0

Author

Jaume Oliver Lafont, Oct 03 2009

Keywords

• frac
• nonn
• easy

Comments

Numerators are all 1.

Setting x=1/3 into 1/(3*x)*log((1+x)/(1-x)^2) = Sum_{k>=0} x^k/((2-(-1)^k)*(k+1)),

log(3) = Sum_{k>=0} 1/((2-(-1)^k)*(k+1)*3^k) = Sum_{k>=0} (9/(2k+1)+1/(2k+2))/9^(k+1) is obtained.

It appears that this is also the first differences of the generalized decagonal numbers A074377. - Omar E. Pol, Sep 10 2011

It appears that this is also A005408 and positive terms of A008588 interleaved. - Omar E. Pol, May 28 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1004
• Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A005408, A008588, A074377, A154920.

Programs

Magma

[(2-(-1)^n)*(n+1): n in [0..350]]; // Vincenzo Librandi, Apr 04 2011

Mathematica

LinearRecurrence[{0,2,0,-1}, {1, 6, 3, 12}, 50] (* Vincenzo Librandi, Feb 22 2012 *)

PARI

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

Formula

G.f.: (1+6*x+x^2)/(1-x^2)^2.

a(n) = (2-(-1)^n)*(n+1) (see PARI's code by Jaume Oliver Lafont).

a(2n)= 2n+1. a(2n+1) = 6*(n+1). - R. J. Mathar, Apr 02 2011

With offset 1 this sequence is multiplicative (in fact, a generalized totient function): a(p^e) = p^e for any odd prime p and a(2^e) = 3*2^e for e >= 1. - Charles R Greathouse IV, Mar 09 2015

With offset 1, Dirichlet g.f.: zeta(s-1) * (1 + 2^(2-s)). - Amiram Eldar, Oct 25 2023