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 A033579 #100 Feb 16 2025 08:32:36
%S A033579 0,4,20,48,88,140,204,280,368,468,580,704,840,988,1148,1320,1504,1700,
%T A033579 1908,2128,2360,2604,2860,3128,3408,3700,4004,4320,4648,4988,5340,
%U A033579 5704,6080,6468,6868,7280,7704,8140,8588,9048,9520,10004,10500,11008,11528,12060
%N A033579 Four times pentagonal numbers: a(n) = 2*n*(3*n-1).
%C A033579 Subsequence of A062717: A010052(6*a(n)+1) = 1. - _Reinhard Zumkeller_, Feb 21 2011
%C A033579 Sequence found by reading the line from 0, in the direction 0, 4, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - _Omar E. Pol_, Sep 08 2011
%H A033579 Ivan Panchenko, <a href="/A033579/b033579.txt">Table of n, a(n) for n = 0..1000</a>
%H A033579 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>.
%H A033579 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pentagonal_number">Pentagonal number</a>.
%H A033579 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A033579 a(n) = 4*n*(3*n-1)/2 = 6*n^2 - 2*n = 4*A000326(n). - _Omar E. Pol_, Dec 11 2008
%F A033579 a(n) = 2*A049450(n). - _Omar E. Pol_, Dec 13 2008
%F A033579 a(n) = a(n-1) + 12*n - 8 for n > 0, a(0)=0. - _Vincenzo Librandi_, Aug 05 2010
%F A033579 a(n) = A014642(n)/2. - _Omar E. Pol_, Aug 19 2011
%F A033579 G.f.: x*(4+8*x)/(1-3*x+3*x^2-x^3). - _Colin Barker_, Jan 06 2012
%F A033579 a(n) = A191967(2*n). - _Reinhard Zumkeller_, Jul 07 2012
%F A033579 a(n) = A181617(n+1) - A181617(n). - _J. M. Bergot_, Jun 28 2013
%F A033579 a(n) = (A174371(n) - 1)/6. - _Miquel Cerda_, Jul 28 2016
%F A033579 From _Ilya Gutkovskiy_, Jul 28 2016: (Start)
%F A033579 E.g.f.: 2*x*(2 + 3*x)*exp(x).
%F A033579 a(n+1) = Sum_{k=0..n} A017569(k).
%F A033579 Sum_{i>0} 1/a(i) = (9*log(3) - sqrt(3)*Pi)/12 = 0.3705093754425278... (End)
%F A033579 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(2*sqrt(3)) - log(2). - _Amiram Eldar_, Feb 20 2022
%p A033579 seq(4*binomial(3*n,2)/3, n=0..45); # _G. C. Greubel_, Oct 09 2019
%t A033579 4 PolygonalNumber[5, Range[0, 45]] (* _Michael De Vlieger_, Aug 02 2016, Version 10.4 *)
%o A033579 (PARI) a(n)=2*n*(3*n-1) \\ _Charles R Greathouse IV_, Jun 28 2013
%o A033579 (Magma) [4*Binomial(3*n,2)/3: n in [0..45]]; // _G. C. Greubel_, Oct 09 2019
%o A033579 (Sage) [4*binomial(3*n,2)/3 for n in (0..45)] # _G. C. Greubel_, Oct 09 2019
%o A033579 (GAP) List([0..45], n-> 4*Binomial(3*n,2)/3 ); # _G. C. Greubel_, Oct 09 2019
%Y A033579 Cf. A000326, A001318, A014642, A033579, A033580, A033581, A049450, A186423.
%K A033579 nonn,easy
%O A033579 0,2
%A A033579 _N. J. A. Sloane_
%E A033579 More terms from _Michel Marcus_, Mar 04 2014