A154516 a(n) = 9n^2 - n.

8, 34, 78, 140, 220, 318, 434, 568, 720, 890, 1078, 1284, 1508, 1750, 2010, 2288, 2584, 2898, 3230, 3580, 3948, 4334, 4738, 5160, 5600, 6058, 6534, 7028, 7540, 8070, 8618, 9184, 9768, 10370, 10990, 11628, 12284, 12958, 13650, 14360

Offset

1,1

Comments

The identity (648*n^2-72*n+1)^2-(9*n^2-n)*(216*n-12)^2=1 can be written as A154514(n)^2-a(n)*A154518(n)^2=1 (see also the second comment in A154514). - Vincenzo Librandi, Jan 30 2012

The continued fraction expansion of sqrt(a(n)) is [3n-1; {1, 4, 1, 6n-2}]. For n=1, this collapses to [2; {1, 4}]. - Magus K. Chu, Sep 06 2022

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Jan 30 2012

G.f.: x*(-8-10*x)/(x-1)^3. - Vincenzo Librandi, Jan 30 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {8, 34, 78}, 50] (* Vincenzo Librandi, Jan 30 2012 *)

Prog

(Sage) [lucas_number1(3, 3*n, n) for n in range(0, 41)] # Zerinvary Lajos, Nov 20 2009
(PARI) a(n)=9*n^2-n \\ Charles R Greathouse IV, Dec 27 2011
(Magma) I:=[8, 34, 78]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jan 30 2012

Crossrefs

Cf. A154514, A154518.

Sequence in context: A373356 A307091 A024847 * A212744 A298174 A249743

Adjacent sequences: A154513 A154514 A154515 * A154517 A154518 A154519

Keyword

nonn,easy

Author

Vincenzo Librandi, Jan 11 2009

Status

approved