This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A165998 #55 Jun 02 2025 02:05:47
%S A165998 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 A165998 78,27,84,29,90,31,96,33,102,35,108,37,114,39,120,41,126,43,132,45,
%U A165998 138,47,144,49,150,51,156,53,162,55,168,57,174,59,180,61,186,63,192,65,198
%N A165998 Denominators of Taylor series expansion of 1/(3*x)*log((1+x)/(1-x)^2).
%C A165998 Numerators are all 1.
%C A165998 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 A165998 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 A165998 It appears that this is also the first differences of the generalized decagonal numbers A074377. - _Omar E. Pol_, Sep 10 2011
%C A165998 It appears that this is also A005408 and positive terms of A008588 interleaved. - _Omar E. Pol_, May 28 2012
%H A165998 Vincenzo Librandi, <a href="/A165998/b165998.txt">Table of n, a(n) for n = 0..1004</a>
%H A165998 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 A165998 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A165998 G.f.: (1+6*x+x^2)/(1-x^2)^2.
%F A165998 a(n) = (2-(-1)^n)*(n+1) (see PARI's code by Jaume Oliver Lafont).
%F A165998 a(2n)= 2n+1. a(2n+1) = 6*(n+1). - _R. J. Mathar_, Apr 02 2011
%F A165998 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 A165998 With offset 1, Dirichlet g.f.: zeta(s-1) * (1 + 2^(2-s)). - _Amiram Eldar_, Oct 25 2023
%t A165998 LinearRecurrence[{0,2,0,-1}, {1, 6, 3, 12}, 50] (* _Vincenzo Librandi_, Feb 22 2012 *)
%o A165998 (PARI) a(n)=(2-(-1)^n)*(n+1)
%o A165998 (Magma) [(2-(-1)^n)*(n+1): n in [0..350]]; // _Vincenzo Librandi_, Apr 04 2011
%Y A165998 Cf. A005408, A008588, A074377, A154920.
%K A165998 frac,nonn,easy
%O A165998 0,2
%A A165998 _Jaume Oliver Lafont_, Oct 03 2009