A033594 a(n) = (n-1)*(2*n-1)*(3*n-1).

-1, 0, 15, 80, 231, 504, 935, 1560, 2415, 3536, 4959, 6720, 8855, 11400, 14391, 17864, 21855, 26400, 31535, 37296, 43719, 50840, 58695, 67320, 76751, 87024, 98175, 110240, 123255, 137256, 152279, 168360

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

Comments

The sequence of n such that n is prime and (2*n+1) is prime is the sequence of Sophie Germain primes A005384 and the subsequence of those for which in addition (3*n+2) is prime is A067256. - Jonathan Vos Post, Dec 15 2004

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A005384, A016921, A067256.

Programs

GAP

List([0..40], n-> (n-1)*(2*n-1)*(3*n-1) ); # G. C. Greubel, Mar 05 2020

Magma

[(n-1)*(2*n-1)*(3*n-1): n in [0..40]]; // Vincenzo Librandi, May 24 2011

Maple

A033594:=n->(n-1)*(2*n-1)*(3*n-1); seq(A033594(n), n=0..40); # Wesley Ivan Hurt, Feb 24 2014

Mathematica

Table[(n-1)*(2*n-1)*(3*n-1),{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 28 2010 *)
LinearRecurrence[{4,-6,4,-1},{-1,0,15,80},40] (* Harvey P. Dale, Aug 23 2012 *)

PARI

vector(41, n, my(m=n-1); (m-1)*(2*m-1)*(3*m-1) ) \\ G. C. Greubel, Mar 05 2020

Sage

[-1]+[n^3*rising_factorial((n-1)/n, 3) for n in (1..40)] # G. C. Greubel, Mar 05 2020

Formula

a(n)*A016921(n) + 1 = A051866(n)^2. - Bruno Berselli, May 23 2011

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(0)=-1, a(1)=0, a(2)=15, a(3)=80. - Harvey P. Dale, Aug 23 2012

G.f.: (-1 +4*x +9*x^2 +24*x^3)/(1-x)^4. - R. J. Mathar, Feb 06 2017

E.g.f.: (-1 + x + 7*x^2 + 6*x^3)*exp(x). - G. C. Greubel, Mar 05 2020

From Amiram Eldar, Jan 03 2021: (Start)

Sum_{n>=2} 1/a(n) = (7 - sqrt(3)*Pi - 16*log(2) + 9*log(3))/4.

Sum_{n>=2} (-1)^n/a(n) = Pi - 7/4 - sqrt(3)*Pi/2 + 2*log(2). (End)