A013656 a(n) = n*(9*n-2).

0, 7, 32, 75, 136, 215, 312, 427, 560, 711, 880, 1067, 1272, 1495, 1736, 1995, 2272, 2567, 2880, 3211, 3560, 3927, 4312, 4715, 5136, 5575, 6032, 6507, 7000, 7511, 8040, 8587, 9152, 9735, 10336, 10955, 11592, 12247, 12920, 13611, 14320, 15047, 15792, 16555

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

For n>0, numbers such that sqrt(a(n)) has the continued fraction {k;[1,1,1,2k]}, where the part in [] is repeated and k is of the form 3m+2 (A016789). - Bruno Berselli, May 30 2013

For n >= 1, the continued fraction expansion of sqrt(4*a(n)) is [6n-1; {3, 3n-1, 3, 12n-2}]. - Magus K. Chu, Sep 18 2022

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A010701, A017257, A185019.

Cf. A016789, A061039, A144454.

Programs

Magma

[n*(9*n-2): n in [0..60]]; // G. C. Greubel, Mar 11 2022

Mathematica

Table[n(9n-2),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,7,32},50] (* Harvey P. Dale, Jul 07 2012 *)

PARI

a(n)=n*(9*n-2) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n+1) = A144454(9*n+7) = A061039(27*n+21). - Paul Curtz, Nov 05 2008

a(n) = a(n-1) + 18*n - 11 with n>0, a(0)=0. - Vincenzo Librandi, Nov 22 2010

a(0)=0, a(1)=7, a(2)=32, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jul 07 2012

From G. C. Greubel, Mar 11 2022: (Start)

G.f.: x*(7 - 11*x)/(1-x)^3.

E.g.f.: x*(7 + 9*x)*exp(x). (End)

Sum_{n>=1} 1/a(n) = -(psi(7/9)+gamma)/2 = (A354640-A001620)/2 = 0.22000753... - R. J. Mathar, Apr 22 2024