A139570 a(n) = 2*n*(n+3).

0, 8, 20, 36, 56, 80, 108, 140, 176, 216, 260, 308, 360, 416, 476, 540, 608, 680, 756, 836, 920, 1008, 1100, 1196, 1296, 1400, 1508, 1620, 1736, 1856, 1980, 2108, 2240, 2376, 2516, 2660, 2808, 2960, 3116, 3276, 3440, 3608, 3780, 3956, 4136, 4320, 4508, 4700, 4896

Offset: 0

Author

Omar E. Pol, May 19 2008

Keywords

• easy
• nonn

Comments

Numbers n such that 2*n + 9 is a square. - Vincenzo Librandi, Nov 24 2010

a(n) appears also as the fourth member of the quartet [p0(n), p1(n), p2(n), a(n)] of the square of [n, n+1, n+2, n+3] in the Clifford algebra Cl_2 for n >= 0. p0(n) = -A147973(n+3), p1(n) = A046092(n), and p2(n) = A054000(n+1). See a comment on A147973, also with a reference. - Wolfdieter Lang, Oct 15 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000096, A001105, A020739, A022998, A028552, A046092, A054000, A067728, A147973.

Programs

Mathematica

CoefficientList[Series[4 x (2 - x)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, May 23 2014 *)

PARI

a(n)=2*n*(n+3) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 2*A028552(n) = 2*n^2 + 6*n = n*(2*n+6).

a(n) = a(n-1) + 4*n + 4 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010

From Paul Curtz, Mar 27 2011: (Start)

a(n) = A022998(n)*A022998(n+3).

a(n) = 4*A000096(n). (End)

G.f.: 4*x*(2 - x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 31 2011

From Amiram Eldar, Dec 23 2022: (Start)

Sum_{n>=1} 1/a(n) = 11/36.

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/3 - 5/36. (End)

From Elmo R. Oliveira, Nov 16 2024: (Start)

E.g.f.: 2*exp(x)*x*(4 + x).

a(n) = n*A020739(n).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)