A085001 a(n) = (3*n+1)*(3*n+4).

4, 28, 70, 130, 208, 304, 418, 550, 700, 868, 1054, 1258, 1480, 1720, 1978, 2254, 2548, 2860, 3190, 3538, 3904, 4288, 4690, 5110, 5548, 6004, 6478, 6970, 7480, 8008, 8554, 9118, 9700, 10300, 10918, 11554, 12208, 12880, 13570, 14278, 15004, 15748, 16510, 17290, 18088

Offset: 0

Author

Gary W. Adamson, Jun 17 2003

Keywords

• nonn
• easy

References

• L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 38.

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. A145910.

Programs

Magma

[(3*n+1)*(3*n+4): n in [0..50]]; // Vincenzo Librandi, Jul 08 2012

Mathematica

CoefficientList[Series[2*(2+8x-x^2)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)
Table[(3n+1)(3n+4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{4,28,70},50] (* Harvey P. Dale, Apr 07 2019 *)

PARI

a(n)=(3*n+1)*(3*n+4) \\ Charles R Greathouse IV, Jun 17 2017

Formula

Sum_{k=0..n} 3/a(k) = 3*(n+1)/(3*n+4). [Corrected by Gary Detlefs, Mar 14 2018]

Sum_{k>=0} 3/a(k) = 1.

From Gary W. Adamson, Jan 03 2007: (Start)

Sum_{k>=0} 1/a(k) = 1/3.

Sum_{k=0..n} 1/a(k) = (n+1)/(3*n+4) [Jolley]. (End) [Corrected by Gary Detlefs, Mar 14 2018]

G.f.: 2*(2+8*x-x^2)/(1-x)^3. - R. J. Mathar, Sep 17 2008

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 08 2012

Sum_{n>=0} (-1)^n/a(n) = 2*Pi/(9*sqrt(3)) + 2*log(2)/9 - 1/3. - Amiram Eldar, Oct 08 2023

From Elmo R. Oliveira, Nov 15 2024: (Start)

E.g.f.: exp(x)*(4 + 24*x + 9*x^2).

a(n) = 2*A145910(n). (End)

Extensions

Edited by Don Reble, Nov 13 2005