A177099 a(n) = 81*n^2 + 2*n.

83, 328, 735, 1304, 2035, 2928, 3983, 5200, 6579, 8120, 9823, 11688, 13715, 15904, 18255, 20768, 23443, 26280, 29279, 32440, 35763, 39248, 42895, 46704, 50675, 54808, 59103, 63560, 68179, 72960, 77903, 83008, 88275, 93704, 99295, 105048, 110963, 117040, 123279, 129680, 136243, 142968, 149855, 156904, 164115

Offset

1,1

Comments

The identity (81*n + 1)^2 - (81*n^2 + 2*n)*9^2 = 1 can be written as A158123(n)^2 - a(n)*9^2 = 1 (see Barbeau's paper in link). - Vincenzo Librandi, Feb 03 2012

Also, the identity (13122*n^2 + 324*n + 1)^2 - (81*n^2 + 2*n)*(1458*n + 18)^2 = 1 can be written as A157506(n)^2 - a(n)*A157505(n)^2 = 1. - Vincenzo Librandi, Feb 04 2012

This last formula is the case s=9 of the identity (2*s^4*n^2 + 4*s^2*n + 1)^2 - (s^2*n^2 + 2*n)*(2*s^3*n + 2*s)^2 = 1. - Bruno Berselli, Feb 04 2011

Links

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

Vincenzo Librandi, X^2-AY^2=1

E. J. Barbeau, Polynomial Excursions, Chapter 10: Diophantine equations (2010), pages 84-85 (row 15 in the first table at p. 85, case d(t) = t*(9^2*t+2)).

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

Formula

G.f.: x*(-79*x - 83)/(x - 1)^3. - Harvey P. Dale, Mar 23 2011

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 03 2012

Mathematica

Table[81n^2+2n, {n, 50}]  (* Harvey P. Dale, Mar 23 2011 *)
LinearRecurrence[{3, -3, 1}, {83, 328, 735}, 40] (* Vincenzo Librandi, Feb 03 2012 *)

Prog

(Magma) [ 81*n^2+2*n: n in [1..50] ];
(PARI) for(n=1, 50, print1(81*n^2+2*n", ")); \\ Vincenzo Librandi, Feb 03 2012

Crossrefs

Cf. A157505, A157506, A158123.

Sequence in context: A288172 A341338 A031433 * A327850 A061525 A293979

Adjacent sequences: A177096 A177097 A177098 * A177100 A177101 A177102

Keyword

nonn,easy

Author

Vincenzo Librandi, Nov 18 2010

Extensions

G.f. adapted to the offset by Bruno Berselli, Apr 01 2011

Status

approved