A098247 First differences of Chebyshev polynomials S(n,227)=A098245(n) with Diophantine property.

1, 226, 51301, 11645101, 2643386626, 600037119001, 136205782626601, 30918112619119426, 7018275358757483101, 1593117588325329544501, 361630674274491049118626, 82088569942721142820383601

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

(15*b(n))^2 - 229*a(n)^2 = -4 with b(n)=A098246(n) give all positive solutions of this Pell equation.

Examples

			All positive solutions of Pell equation x^2 - 229*y^2 = -4 are (15=15*1,1), (3420=15*228,226), (776325=15*51755,51301), (176222355=15*11748157,11645101), ...
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..423
• 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 (227,-1).
• Index entries for sequences related to Chebyshev polynomials.

Programs

GAP

a:=[1,226];; for n in [3..20] do a[n]:=227*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Magma

I:=[1,226]; [n le 2 select I[n] else 227*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

Mathematica

LinearRecurrence[{227,-1}, {1,226}, 20] (* G. C. Greubel, Aug 01 2019 *)

PARI

my(x='x+O('x^20)); Vec((1-x)/(1-227*x+x^2)) \\ G. C. Greubel, Aug 01 2019

Sage

((1-x)/(1-227*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

Formula

a(n) = S(n, 227) - S(n-1, 227) = T(2*n+1, sqrt(229)/2)/(sqrt(229)/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 second kind, A053120.

a(n) = ((-1)^n)*S(2*n, 15*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.

G.f.: (1-x)/(1-227*x+x^2).

a(n) = 227*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=226. - Philippe Deléham, Nov 18 2008