OFFSET
0,2
COMMENTS
(23*a(n))^2 - 533*b(n)^2 = -4 with b(n)=A098259(n) give all positive solutions of this Pell equation.
LINKS
Muniru A Asiru, Table of n, a(n) for n = 0..300
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for linear recurrences with constant coefficients, signature (531, -1).
FORMULA
a(n) = S(n, 531) + S(n-1, 531) = S(2*n, sqrt(533)), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x). S(n, 531)=A098257(n).
a(n)= (-2/23)*i*((-1)^n)*T(2*n+1, 23*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-531*x+x^2).
a(n) = 531*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=532. - Philippe Deléham, Nov 18 2008
EXAMPLE
All positive solutions of Pell equation x^2 - 533*y^2 = -4 are (23=23*1,1), (12236=23*532,530), (6497293=23*282491,281429), (3450050347=23*150002189,149438269), ...
MATHEMATICA
LinearRecurrence[{531, -1}, {1, 532}, 20] (* Harvey P. Dale, Oct 09 2018 *)
PROG
(GAP) a:=[1, 532];; for n in [3..12] do a[n]:=531*a[n-1]-a[n-2]; od; Print(a); # Muniru A Asiru, Apr 29 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 10 2004
STATUS
approved