A156676 a(n) = 81*n^2 - 44*n + 6.

6, 43, 242, 603, 1126, 1811, 2658, 3667, 4838, 6171, 7666, 9323, 11142, 13123, 15266, 17571, 20038, 22667, 25458, 28411, 31526, 34803, 38242, 41843, 45606, 49531, 53618, 57867, 62278, 66851, 71586, 76483, 81542, 86763, 92146, 97691, 103398, 109267, 115298, 121491

Offset: 0

Author

Vincenzo Librandi, Feb 15 2009

Keywords

• nonn
• easy

Comments

The identity (6561*n^2 - 3564*n + 485)^2 - (81*n^2 - 44*n + 6)*(729*n - 198)^2 = 1 can be written as A156774(n)^2 - a(n)*A156772(n)^2 = 1 for n > 0.

For n >= 1, the continued fraction expansion of sqrt(a(n)) is [9n-3; {1, 1, 3, 1, 9n-4, 1, 3, 1, 1, 18n-6}]. - Magus K. Chu, Sep 13 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000290, A017137, A017245, A156772, A156774.

Programs

Magma

[81*n^2 - 44*n + 6: n in [0..40] ];

Maple

A156676:=n->81*n^2-44*n+6: seq(A156676(n), n=0..100); # Wesley Ivan Hurt, Apr 26 2017

Mathematica

LinearRecurrence[{3,-3,1},{6,43,242},40]
Table[81n^2-44n+6,{n,0,40}] (* Harvey P. Dale, Oct 29 2019 *)

PARI

a(n)=81*n^2-44*n+6 \\ Charles R Greathouse IV, Dec 23 2011

Formula

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

G.f.: (6 + 25*x + 131*x^2)/(1-x)^3.

a(n) = A000290(A017245(n-1)) - A017137(n-1). - Reinhard Zumkeller, Jul 13 2010

E.g.f.: (6 + 37*x + 81*x^2)*exp(x). - Elmo R. Oliveira, Oct 19 2024

Extensions

Edited by Charles R Greathouse IV, Jul 25 2010