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 A157667(n)^2 - a(n)*A157666(n)^2 = 1.
The continued fraction expansion of sqrt(a(n)) is [27*n-19; {5, 2, 1, 1, 27*n-20, 1, 1, 2, 5, 54*n-38}]. - Magus K. Chu, Nov 20 2022
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1 [Dead link]
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*(67 + 1037*x + 354*x^2)/(1-x)^3.
E.g.f.: (1 - 287*x + 729*x^2)*exp(x) - 354. - G. C. Greubel, Nov 17 2018
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {67, 1238, 3867}, 40]
PROG
(Magma) I:=[67, 1238, 3867]; [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 - 1016*n + 354.
(Sage) [729*n^2 - 1016*n + 354 for n in (1..40)] # G. C. Greubel, Nov 17 2018
(GAP) List([1..40], n -> 729*n^2 - 1016*n + 354); # G. C. Greubel, Nov 17 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Mar 04 2009
STATUS
approved