A078988 Chebyshev sequence with Diophantine property.

1, 65, 4289, 283009, 18674305, 1232221121, 81307919681, 5365090477825, 354014663616769, 23359602708228929, 1541379764079492545, 101707704826538279041, 6711167138787446924161, 442835323455144958715585, 29220420180900779828304449, 1928104896615996323709378049

Offset

0,2

Comments

Bisection (even part) of A041025.

(4*A078989(n))^2 - 17*a(n)^2 = -1 (Pell -1 equation, see A077232-3).

Starting with a(1), hypotenuses of primitive Pythagorean triples in A195619 and A195620. - Clark Kimberling, Sep 22 2011

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 sequences related to Chebyshev polynomials.

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

Row 66 of array A094954.

Cf. A097316 for S(n, 66).

Row 4 of array A188647.

Sequence in context: A188772 A207186 A189062 * A027535 A110900 A084272

Adjacent sequences: A078985 A078986 A078987 * A078989 A078990 A078991

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 10 2003

Status

approved