%I #53 May 20 2023 14:38:29
%S 0,6,28,66,120,190,276,378,496,630,780,946,1128,1326,1540,1770,2016,
%T 2278,2556,2850,3160,3486,3828,4186,4560,4950,5356,5778,6216,6670,
%U 7140,7626,8128,8646,9180,9730,10296,10878,11476,12090,12720,13366,14028,14706
%N a(n) = 2*n*(4*n - 1).
%C Even hexagonal numbers.
%C 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 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 Vincenzo Librandi, <a href="/A014635/b014635.txt">Table of n, a(n) for n = 0..880</a>
%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H 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 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 Leo Tavares, <a href="/A014635/a014635.jpg">Illustration: Diamond Cut Hexagons</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = C(4*n,2), n>=0. - _Zerinvary Lajos_, Jan 02 2007
%F O.g.f.: 2*x*(3+5*x)/(1-x)^3. - _R. J. Mathar_, May 06 2008
%F a(n) = 8*n^2 - 2*n. - _Omar E. Pol_, May 07 2008
%F a(n) = a(n-1) + 16*n - 10 (with a(0)=0). - _Vincenzo Librandi_, Nov 20 2010
%F E.g.f.: (8*x^2 + 6*x)*exp(x). - _G. C. Greubel_, Jul 18 2017
%F From _Vaclav Kotesovec_, Aug 18 2018: (Start)
%F Sum_{n>=1} 1/a(n) = 3*log(2)/2 - Pi/4.
%F Sum_{n>=1} (-1)^n / a(n) = log(2)/2 + log(1+sqrt(2))/sqrt(2) - Pi / 2^(3/2). (End)
%F a(n) = A154105(n-1) - A016754(n-1). - _Leo Tavares_, May 02 2023
%p [seq(binomial(4*n,2),n=0..43)]; # _Zerinvary Lajos_, Jan 02 2007
%t s=0;lst={s};Do[s+=n++ +6;AppendTo[lst, s], {n, 0, 7!, 16}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 16 2008 *)
%t Table[2*n*(4*n - 1), {n,0,50}] (* _G. C. Greubel_, Jul 18 2017 *)
%t PolygonalNumber[6,Range[0,90,2]] (* or *) LinearRecurrence[{3,-3,1},{0,6,28},50] (* _Harvey P. Dale_, Jan 21 2023 *)
%o (Magma) [2*n*(4*n-1): n in [0..50]]; // _Vincenzo Librandi_, Apr 25 2011
%o (PARI) a(n)=2*n*(4*n-1) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A000217, A000384, A000396, A134708, A139596.
%Y Cf. A154105, A016754.
%K nonn,easy
%O 0,2
%A _Mohammad K. Azarian_
%E More terms from _Erich Friedman_