A033569 a(n) = (2*n - 1)*(3*n + 1).

-1, 4, 21, 50, 91, 144, 209, 286, 375, 476, 589, 714, 851, 1000, 1161, 1334, 1519, 1716, 1925, 2146, 2379, 2624, 2881, 3150, 3431, 3724, 4029, 4346, 4675, 5016, 5369, 5734, 6111, 6500, 6901, 7314, 7739, 8176, 8625, 9086, 9559, 10044, 10541, 11050, 11571

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

Comments

For n>0, a(n) is the sum of the numbers from 2n+2 to 4n. The last digit of a(n) corresponds to the last digit of the squares mod 10 (A008959). Binomial Transform of a(n) starts: -1, 3, 28, 124, 432, 1328, 3776, 10176, 26368, ... . - Wesley Ivan Hurt, Dec 06 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. A008959, A060747, A016777, A259758 (subsequence).

Programs

GAP

List([0..50], n-> (2*n-1)*(3*n+1)) # G. C. Greubel, Apr 02 2019

Haskell

a033569 n = (2 * n - 1) * (3 * n + 1)
a033569_list = map a033569 [0..]
-- Reinhard Zumkeller, Jul 05 2015

Magma

[(2*n-1)*(3*n+1): n in [0..50]]; // Vincenzo Librandi, Jul 07 2012

Maple

A033569:=n->(2*n-1)*(3*n+1): seq(A033569(n), n=0..50); # Wesley Ivan Hurt, Dec 06 2014

Mathematica

CoefficientList[Series[(-1+7*x+6*x^2)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 07 2012 *)

PARI

a(n)=(2*n-1)*(3*n+1) \\ Charles R Greathouse IV, Jun 17 2017

Sage

[(2*n-1)*(3*n+1) for n in (0..50)] # G. C. Greubel, Apr 02 2019

Formula

G.f.: (-1+7*x+6*x^2)/(1-x)^3. - Vincenzo Librandi, Jul 07 2012

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Jul 07 2012

E.g.f.: (-1+5*x+6*x^2)*e^x. - Robert Israel, Dec 07 2014

a(n) = A060747(n) * A016777(n). - Reinhard Zumkeller, Jul 05 2015

Sum_{n>=0} 1/a(n) = 2/5*(log(2)-1) -sqrt(3)*Pi/30 -3*log(3)/10 = -0.6337047... - R. J. Mathar, Apr 22 2024