A177073 a(n) = (9*n+4)*(9*n+5).

20, 182, 506, 992, 1640, 2450, 3422, 4556, 5852, 7310, 8930, 10712, 12656, 14762, 17030, 19460, 22052, 24806, 27722, 30800, 34040, 37442, 41006, 44732, 48620, 52670, 56882, 61256, 65792, 70490, 75350, 80372, 85556, 90902, 96410, 102080, 107912, 113906, 120062

Offset: 0

Author

Vincenzo Librandi, May 31 2010

Keywords

• nonn
• easy

Comments

Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 81*A002061(n+1) - 61. - Bruno Berselli, Aug 24 2010

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. A002061, A017209, A017221, A177059.

Programs

Magma

[(9*n+4)*(9*n+5): n in [0..50]]; // Vincenzo Librandi, Apr 08 2013

Mathematica

f[n_] := Module[{c = 9n}, (c+4)(c+5)]; Array[f, 40, 0] (* or *) LinearRecurrence[{3, -3, 1}, {20, 182, 506}, 40] (* Harvey P. Dale, Jun 24 2011 *)

PARI

a(n)=(9*n+4)*(9*n+5) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 162*n + a(n-1) with n > 0, a(0)=20.

From Harvey P. Dale, Jun 24 2011: (Start)

a(0)=20, a(1)=182, a(2)=506, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: -2*(x*(10*x+61)+10)/(x-1)^3. (End)

From Amiram Eldar, Feb 19 2023: (Start)

a(n) = A017209(n)*A017221(n).

Sum_{n>=0} 1/a(n) = tan(Pi/18)*Pi/9.

Product_{n>=0} (1 - 1/a(n)) = sec(Pi/18)*cos(sqrt(5)*Pi/18).

Product_{n>=0} (1 + 1/a(n)) = sec(Pi/18)*cosh(sqrt(3)*Pi/18). (End)

E.g.f.: exp(x)*(20 + 81*x*(2 + x)). - Elmo R. Oliveira, Oct 18 2024

Extensions

Edited by N. J. A. Sloane, Jun 22 2010