A014635 a(n) = 2*n*(4*n - 1).

0, 6, 28, 66, 120, 190, 276, 378, 496, 630, 780, 946, 1128, 1326, 1540, 1770, 2016, 2278, 2556, 2850, 3160, 3486, 3828, 4186, 4560, 4950, 5356, 5778, 6216, 6670, 7140, 7626, 8128, 8646, 9180, 9730, 10296, 10878, 11476, 12090, 12720, 13366, 14028, 14706

Offset: 0

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

Comments

Even hexagonal numbers.

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

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..880
• Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
• Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
• Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
• Leo Tavares, Illustration: Diamond Cut Hexagons
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A000384, A000396, A134708, A139596.

Cf. A154105, A016754.

Programs

Magma

[2*n*(4*n-1): n in [0..50]]; // Vincenzo Librandi, Apr 25 2011

Maple

[seq(binomial(4*n,2),n=0..43)]; # Zerinvary Lajos, Jan 02 2007

Mathematica

s=0;lst={s};Do[s+=n++ +6;AppendTo[lst, s], {n, 0, 7!, 16}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
Table[2*n*(4*n - 1), {n,0,50}] (* G. C. Greubel, Jul 18 2017 *)
PolygonalNumber[6,Range[0,90,2]] (* or *) LinearRecurrence[{3,-3,1},{0,6,28},50] (* Harvey P. Dale, Jan 21 2023 *)

PARI

a(n)=2*n*(4*n-1) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = C(4*n,2), n>=0. - Zerinvary Lajos, Jan 02 2007

O.g.f.: 2*x*(3+5*x)/(1-x)^3. - R. J. Mathar, May 06 2008

a(n) = 8*n^2 - 2*n. - Omar E. Pol, May 07 2008

a(n) = a(n-1) + 16*n - 10 (with a(0)=0). - Vincenzo Librandi, Nov 20 2010

E.g.f.: (8*x^2 + 6*x)*exp(x). - G. C. Greubel, Jul 18 2017

From Vaclav Kotesovec, Aug 18 2018: (Start)

Sum_{n>=1} 1/a(n) = 3*log(2)/2 - Pi/4.

Sum_{n>=1} (-1)^n / a(n) = log(2)/2 + log(1+sqrt(2))/sqrt(2) - Pi / 2^(3/2). (End)

a(n) = A154105(n-1) - A016754(n-1). - Leo Tavares, May 02 2023

Extensions

More terms from Erich Friedman