A157667 a(n) = 531441*n^2 - 740664*n + 258065.

48842, 902501, 2819042, 5798465, 9840770, 14945957, 21114026, 28344977, 36638810, 45995525, 56415122, 67897601, 80442962, 94051205, 108722330, 124456337, 141253226, 159112997, 178035650, 198021185, 219069602, 241180901

Offset

1,1

Comments

The identity (531441*n^2 - 740664*n + 258065)^2 - (729*n^2 - 1016*n + 354)*(19683*n - 13716)^2 = 1 can be written as a(n)^2 - A157665(n)*A157666(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*(48842 + 755975*x + 258065*x^2)/(1-x)^3.

E.g.f.: (258065 - 209223*x + 531441*x^2)*exp(x) - 258065. - G. C. Greubel, Nov 17 2018

Mathematica

LinearRecurrence[{3, -3, 1}, {48842, 902501, 2819042}, 40]

Prog

(Magma) I:=[48842, 902501, 2819042]; [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 - 740664*n + 258065.
(Sage) [531441*n^2 - 740664*n + 258065 for n in (1..40)] # G. C. Greubel, Nov 17 2018
(GAP) List([1..40], n -> 531441*n^2 - 740664*n + 258065); # G. C. Greubel, Nov 17 2018

Crossrefs

Cf. A157665, A157666.

Sequence in context: A067869 A175278 A336189 * A176373 A237087 A061737

Adjacent sequences: A157664 A157665 A157666 * A157668 A157669 A157670

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 04 2009

Status

approved