OFFSET
0,2
COMMENTS
(27*b(n))^2 - 733*a(n)^2 = -4 with b(n) = A098291(n) give all positive solutions of this Pell equation.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..340
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 (731,-1).
FORMULA
a(n) = ((-1)^n)*S(2*n, 27*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-731*x+x^2).
a(n) = S(n, 731) - S(n-1, 731) = T(2*n+1, sqrt(733)/2)/(sqrt(733)/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) = 731*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=730. - Philippe Deléham, Nov 18 2008
Sum_{n>=0} 1/(a(n)+1) = sqrt(733)/54. - Amiram Eldar, Jan 01 2026
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), ...
MATHEMATICA
LinearRecurrence[{731, -1}, {1, 730}, 20] (* Harvey P. Dale, Nov 15 2013 *)
PROG
(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-731*x+x^2)) \\ G. C. Greubel, Aug 01 2019
(Magma) I:=[1, 730]; [n le 2 select I[n] else 731*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
(SageMath) ((1-x)/(1-731*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 730];; for n in [3..20] do a[n]:=731*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 10 2004
STATUS
approved