A098244 First differences of Chebyshev polynomials S(n,171)=A097844(n) with Diophantine property.

1, 170, 29069, 4970629, 849948490, 145336221161, 24851643870041, 4249485765555850, 726637214266180309, 124250714153751276989, 21246145483077202184810, 3632966626892047822325521, 621216047053057100415479281, 106224311079445872123224631530

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

(13*b(n))^2 - 173*a(n)^2 = -4 with b(n)=A097845(n) give all positive solutions of this Pell equation.

Examples

			All positive solutions of Pell equation x^2 - 173*y^2 = -4 are (13=13*1,1), (2236=13*172,170), (382343=13*29411,29069), (65378417=13*5029109,4970629), ...
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..446
• 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 (171,-1).
• Index entries for sequences related to Chebyshev polynomials.

Programs

GAP

a:=[1,170];; for n in [3..20] do a[n]:=171*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Magma

I:=[1,170]; [n le 2 select I[n] else 171*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

Mathematica

LinearRecurrence[{171,-1}, {1,170}, 20] (* G. C. Greubel, Aug 01 2019 *)

PARI

my(x='x+O('x^20)); Vec((1-x)/(1-171*x+x^2)) \\ G. C. Greubel, Aug 01 2019

Sage

((1-x)/(1-171*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

Formula

a(n) = ((-1)^n)*S(2*n, 13*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.

G.f.: (1-x)/(1-171*x+x^2).

a(n) = S(n, 171) - S(n-1, 171) = T(2*n+1, sqrt(173)/2)/(sqrt(173)/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) = 171*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=170. - Philippe Deléham, Nov 18 2008