A165998 - OEIS

%I #53 Oct 25 2023 09:30:16

%S 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,

%T 78,27,84,29,90,31,96,33,102,35,108,37,114,39,120,41,126,43,132,45,

%U 138,47,144,49,150,51,156,53,162,55,168,57,174,59,180,61,186,63,192,65,198

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

%C Numerators are all 1.

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

%C 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.

%C It appears that this is also the first differences of the generalized decagonal numbers A074377. - _Omar E. Pol_, Sep 10 2011

%C It appears that this is also A005408 and positive terms of A008588 interleaved. - _Omar E. Pol_, May 28 2012

%H Vincenzo Librandi, <a href="/A165998/b165998.txt">Table of n, a(n) for n = 0..1004</a>

%H Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, <a href="https://doi.org/10.3934/era.2020057">Recursive sequences and Girard-Waring identities with applications in sequence transformation</a>, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).

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

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

%F a(2n)= 2n+1. a(2n+1) = 6*(n+1). - _R. J. Mathar_, Apr 02 2011

%F 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

%F With offset 1, Dirichlet g.f.: zeta(s-1) * (1 + 2^(2-s)). - _Amiram Eldar_, Oct 25 2023

%t LinearRecurrence[{0,2,0,-1}, {1, 6, 3, 12}, 50] (* _Vincenzo Librandi_, Feb 22 2012 *)

%o (PARI) a(n)=(2-(-1)^n)*(n+1)

%o (Magma) [(2-(-1)^n)*(n+1): n in [0..350]]; // _Vincenzo Librandi_, Apr 04 2011

%Y Cf. A005408, A008588, A074377, A154920.

%K frac,nonn,easy

%O 0,2

%A _Jaume Oliver Lafont_, Oct 03 2009