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

0,2

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).

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

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 *)

Prog

(Magma) [2*n*(4*n-1): n in [0..50]]; // Vincenzo Librandi, Apr 25 2011
(PARI) a(n)=2*n*(4*n-1) \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

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

Cf. A154105, A016754.

Sequence in context: A081537 A058007 A033588 * A227970 A034955 A117978

Adjacent sequences: A014632 A014633 A014634 * A014636 A014637 A014638

Keyword

nonn,easy

Author

Mohammad K. Azarian

Extensions

More terms from Erich Friedman

Status

approved