OFFSET
0,2
COMMENTS
(27*a(n))^2 - 733*b(n)^2 = -4 with b(n)=A098292(n) give all positive solutions of this Pell equation.
LINKS
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 (731,-1).
FORMULA
a(n) = S(n, 731) + S(n-1, 731) = S(2*n, sqrt(733)), 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, 731)=A098263(n).
a(n) = (-2/27)*i*((-1)^n)*T(2*n+1, 27*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangleA053120.
G.f.: (1+x)/(1-731*x+x^2).
EXAMPLE
All positive solutions of Pell equation x^2 - 733*y^2 = -4 are (27=27*1,1), (19764=27*732,730), (14447457=27*535091,533629), (10561071303=27*391150789,390082069),...
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang,Sep 10 2004
STATUS
approved