A157670 a(n) = 531441*n^2 - 322218*n + 48842.

258065, 1530170, 3865157, 7263026, 11723777, 17247410, 23833925, 31483322, 40195601, 49970762, 60808805, 72709730, 85673537, 99700226, 114789797, 130942250, 148157585, 166435802, 185776901, 206180882, 227647745, 250177490

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 a(n)^2 - A157668(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*(258065 + 755975*x + 48842*x^2)/(1-x)^3.

E.g.f.: (48842 + 209223*x + 531441*x^2)*exp(x) - 48842. - G. C. Greubel, Nov 17 2018

Mathematica

LinearRecurrence[{3, -3, 1}, {258065, 1530170, 3865157}, 40]

Prog

(Magma) I:=[258065, 1530170, 3865157]; [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) = 531441*n^2 - 322218*n + 48842.
(Sage) [531441*n^2 - 322218*n + 48842 for n in (1..40)] # G. C. Greubel, Nov 17 2018
(GAP) List([1..40], n -> 531441*n^2 - 322218*n + 48842); # G. C. Greubel, Nov 17 2018

Crossrefs

Cf. A157668, A157669.

Sequence in context: A251165 A065794 A206253 * A252921 A216204 A153980

Adjacent sequences: A157667 A157668 A157669 * A157671 A157672 A157673

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 04 2009

Status

approved