A098250 First differences of Chebyshev polynomials S(n,291)=A098248(n) with Diophantine property.

1, 290, 84389, 24556909, 7145976130, 2079454496921, 605114112627881, 176086127320216450, 51240457936070359069, 14910797173269154272629, 4338990736963387822975970, 1262631393659172587331734641, 367421396564082259525711804561, 106918363768754278349394803392610

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

(17*b(n))^2 - 293*a(n)^2 = -4 with b(n)=A098249(n) give all positive solutions of this Pell equation.

Examples

			All positive solutions of Pell equation x^2 - 293*y^2 = -4 are (17=17*1,1), (4964=17*292,290), (1444507=17*84971,84389), (420346573=17*24726269,24556909), ...
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..405
• 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 (291,-1).
• Index entries for sequences related to Chebyshev polynomials.

Programs

GAP

a:=[1,290];; for n in [3..20] do a[n]:=291*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Magma

I:=[1,290]; [n le 2 select I[n] else 291*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

Mathematica

LinearRecurrence[{291,-1}, {1,290}, 20] (* G. C. Greubel, Aug 01 2019 *)

PARI

my(x='x+O('x^20)); Vec((1-x)/(1-291*x+x^2)) \\ G. C. Greubel, Aug 01 2019

Sage

((1-x)/(1-291*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

Formula

a(n) = ((-1)^n)*S(2*n, 17*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.

G.f.: (1-x)/(1-291*x+x^2).

a(n) = S(n, 291) - S(n-1, 291) = T(2*n+1, sqrt(293)/2)/(sqrt(293)/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) = 291*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=290. - Philippe Deléham, Nov 18 2008