OFFSET
0,2
COMMENTS
LINKS
Colin Barker, Table of n, a(n) for n = 0..549
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.
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 (66,-1).
FORMULA
G.f.: (1-x)/(1-66*x+x^2).
a(n) = T(2*n+1, sqrt(17))/sqrt(17) = ((-1)^n)*S(2*n, 8*i) = S(n, 66) - S(n-1, 66) with i^2=-1 and T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310.
a(n) = A041025(2*n).
a(n) = 66*a(n-1) - a(n-2) for n>1 ; a(0)=1, a(1)=65. - Philippe Deléham, Nov 18 2008
EXAMPLE
(x,y) = (4,1), (268,65), (17684,4289), ... give the positive integer solutions to x^2 - 17*y^2 =-1.
MATHEMATICA
CoefficientList[Series[(1-x)/(1-66x+x^2), {x, 0, 20}], x] (* Michael De Vlieger, Apr 15 2019 *)
LinearRecurrence[{66, -1}, {1, 65}, 21] (* G. C. Greubel, Aug 01 2019 *)
PROG
(PARI) Vec((1-x)/(1-66*x+x^2) + O(x^20)) \\ Colin Barker, Jun 15 2015
(Magma) I:=[1, 65]; [n le 2 select I[n] else 66*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
(Sage) ((1-x)/(1-66*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 65];; for n in [3..20] do a[n]:=66*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jan 10 2003
STATUS
approved