OFFSET
1,2
COMMENTS
This sequence is part of a solution of a more general problem involving two equations, three sequences a(n), b(n), c(n) and a constant A:
A * c(n)+1 = a(n)^2,
(A+1) * c(n)+1 = b(n)^2, for details see comment in A157014.
This sequence is the c(n) sequence for A=9.
LINKS
Colin Barker, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (1443,-1443,1).
FORMULA
8*a(n)+1 = A097315(n-1)^2.
9*a(n)+1 = A097314(n-1)^2.
G.f.: 152*x^2/(-x^3+1443*x^2-1443*x+1).
a(1) = 0, a(2) = 152, a(3) = 1443*a(2), a(n) = 1443 * (a(n-1)-a(n-2)) + a(n-3) for n > 3.
a(n) = -((721+228*sqrt(10))^(-n)*(-1+(721+228*sqrt(10))^n)*(19+6*sqrt(10)+(-19+6*sqrt(10))*(721+228*sqrt(10))^n))/360. - Colin Barker, Jul 25 2016
Sum_{n>=2} 1/a(n) = (19 - 6*sqrt(10))/4. - Amiram Eldar, Jan 29 2026
MATHEMATICA
LinearRecurrence[{1443, -1443, 1}, {0, 152, 219336}, 20] (* Harvey P. Dale, Jul 18 2019 *)
PROG
(PARI) concat(0, Vec(152*x^2/(-x^3+1443*x^2-1443*x+1) + O(x^20))) \\ Charles R Greathouse IV, Sep 26 2012
(PARI) a(n) = round(-((721+228*sqrt(10))^(-n)*(-1+(721+228*sqrt(10))^n)*(19+6*sqrt(10)+(-19+6*sqrt(10))*(721+228*sqrt(10))^n))/360) \\ Colin Barker, Jul 25 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Weisenhorn, Mar 08 2009, Jun 25 2009
EXTENSIONS
Edited by Alois P. Heinz, Sep 09 2011
STATUS
approved