OFFSET
0,2
COMMENTS
(5*b(n))^2 - 29*a(n)^2 = -4 with b(n) = A097834(n) give all positive solutions of this Pell equation.
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..697
Tanya Khovanova, Recursive Sequences.
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum, Vol. 19 (2019), 11-16.
Index entries for linear recurrences with constant coefficients, signature (27,-1).
FORMULA
a(n) = ((-1)^n)*S(2*n, 5*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-27*x+x^2).
a(n) = S(n, 27) - S(n-1, 27) = T(2*n+1, sqrt(29)/2)/(sqrt(29)/2), with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x) = 0 = U(-1, x) and T(n, x) Chebyshev's polynomials of the first kind, A053120.
a(n) = 27*a(n-1) - a(n-2), a(0)=1, a(1)=26. - Philippe Deléham, Nov 18 2008
Sum_{n>=0} 1/(a(n)+1) = sqrt(29)/10 = A010484 / 10. - Amiram Eldar, Jan 01 2026
EXAMPLE
All positive solutions of Pell equation x^2 - 29*y^2 = -4 are (5 = 5*1, 1), (140 = 5*28, 26), (3775 = 5*755, 701), (101785 = 5*20357, 18901), ...
MATHEMATICA
LinearRecurrence[{27, -1}, {1, 26}, 30] (* Harvey P. Dale, May 31 2013 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((1-x)/(1-27*x+x^2)) \\ G. C. Greubel, Jan 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(1-27*x+x^2) )); // G. C. Greubel, Jan 12 2019
(SageMath) ((1-x)/(1-27*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019
(GAP) a:=[1, 26];; for n in [3..30] do a[n]:=27*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 12 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 10 2004
STATUS
approved