A157668 a(n) = 729*n^2 - 442*n + 67.

354, 2099, 5302, 9963, 16082, 23659, 32694, 43187, 55138, 68547, 83414, 99739, 117522, 136763, 157462, 179619, 203234, 228307, 254838, 282827, 312274, 343179, 375542, 409363, 444642, 481379, 519574, 559227, 600338, 642907, 686934, 732419

Offset

1,1

Comments

The identity (531441*n^2 - 322218*n + 48842)^2 - (729*n^2 - 442*n + 67)*(19683*n - 5967)^2 = 1 can be written as A157670(n)^2 - a(n)*A157669(n)^2 = 1.

Links

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

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

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).

G.f.: x*(-354 - 1037*x - 67*x^2)/(x-1)^3.

E.g.f.: (67 + 287*x + 729*x^2)*exp(x) - 67. - G. C. Greubel, Nov 17 2018

Mathematica

LinearRecurrence[{3, -3, 1}, {354, 2099, 5302}, 40]
Table[729n^2-442n+67, {n, 40}] (* Harvey P. Dale, Dec 25 2019 *)

Prog

(Magma) I:=[354, 2099, 5302]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
(PARI) a(n) = 729*n^2 - 442*n + 67.
(Sage) [729*n^2 - 442*n + 67 for n in (1..40)] # G. C. Greubel, Nov 17 2018
(GAP) List([1..40], n -> 729*n^2 - 442*n + 67); # G. C. Greubel, Nov 17 2018

Crossrefs

Cf. A157669, A157670.

Sequence in context: A270782 A251127 A176197 * A374696 A250157 A293563

Adjacent sequences: A157665 A157666 A157667 * A157669 A157670 A157671

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 04 2009

Status

approved