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 A014635 #53 May 20 2023 14:38:29
%S A014635 0,6,28,66,120,190,276,378,496,630,780,946,1128,1326,1540,1770,2016,
%T A014635 2278,2556,2850,3160,3486,3828,4186,4560,4950,5356,5778,6216,6670,
%U A014635 7140,7626,8128,8646,9180,9730,10296,10878,11476,12090,12720,13366,14028,14706
%N A014635 a(n) = 2*n*(4*n - 1).
%C A014635 Even hexagonal numbers.
%C A014635 Number of edges in the join of two complete graphs of order 3n and n, K_3n * K_n - _Roberto E. Martinez II_, Jan 07 2002
%C A014635 Bisection of A000384. Also, this sequence arises from reading the line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the triangular numbers A000217. Perfect numbers are members of this sequence because a(A134708(n)) = A000396(n). Also, positive members are a bisection of A139596. - _Omar E. Pol_, May 07 2008
%H A014635 Vincenzo Librandi, <a href="/A014635/b014635.txt">Table of n, a(n) for n = 0..880</a>
%H A014635 Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H A014635 Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].
%H A014635 Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3470205">The groupoid of the Triangular Numbers and the generation of related integer sequences</a>, Politecnico di Torino, Italy (2019).
%H A014635 Leo Tavares, <a href="/A014635/a014635.jpg">Illustration: Diamond Cut Hexagons</a>
%H A014635 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A014635 a(n) = C(4*n,2), n>=0. - _Zerinvary Lajos_, Jan 02 2007
%F A014635 O.g.f.: 2*x*(3+5*x)/(1-x)^3. - _R. J. Mathar_, May 06 2008
%F A014635 a(n) = 8*n^2 - 2*n. - _Omar E. Pol_, May 07 2008
%F A014635 a(n) = a(n-1) + 16*n - 10 (with a(0)=0). - _Vincenzo Librandi_, Nov 20 2010
%F A014635 E.g.f.: (8*x^2 + 6*x)*exp(x). - _G. C. Greubel_, Jul 18 2017
%F A014635 From _Vaclav Kotesovec_, Aug 18 2018: (Start)
%F A014635 Sum_{n>=1} 1/a(n) = 3*log(2)/2 - Pi/4.
%F A014635 Sum_{n>=1} (-1)^n / a(n) = log(2)/2 + log(1+sqrt(2))/sqrt(2) - Pi / 2^(3/2). (End)
%F A014635 a(n) = A154105(n-1) - A016754(n-1). - _Leo Tavares_, May 02 2023
%p A014635 [seq(binomial(4*n,2),n=0..43)]; # _Zerinvary Lajos_, Jan 02 2007
%t A014635 s=0;lst={s};Do[s+=n++ +6;AppendTo[lst, s], {n, 0, 7!, 16}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 16 2008 *)
%t A014635 Table[2*n*(4*n - 1), {n,0,50}] (* _G. C. Greubel_, Jul 18 2017 *)
%t A014635 PolygonalNumber[6,Range[0,90,2]] (* or *) LinearRecurrence[{3,-3,1},{0,6,28},50] (* _Harvey P. Dale_, Jan 21 2023 *)
%o A014635 (Magma) [2*n*(4*n-1): n in [0..50]]; // _Vincenzo Librandi_, Apr 25 2011
%o A014635 (PARI) a(n)=2*n*(4*n-1) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A014635 Cf. A000217, A000384, A000396, A134708, A139596.
%Y A014635 Cf. A154105, A016754.
%K A014635 nonn,easy
%O A014635 0,2
%A A014635 _Mohammad K. Azarian_
%E A014635 More terms from _Erich Friedman_