A078988 Chebyshev sequence with Diophantine property.

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

Offset: 0

Author

Wolfdieter Lang, Jan 10 2003

Keywords

• nonn
• easy

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

Examples

			(x,y) = (4,1), (268,65), (17684,4289), ... give the positive integer solutions to x^2 - 17*y^2 =-1.
		

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).

Crossrefs

Row 66 of array A094954.

Cf. A097316 for S(n, 66).

Row 4 of array A188647.

Programs

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

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

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 *)

PARI

Vec((1-x)/(1-66*x+x^2) + O(x^20)) \\ Colin Barker, Jun 15 2015

Sage

((1-x)/(1-66*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

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