A098262 First differences of Chebyshev polynomials S(n,627)=A098260(n) with Diophantine property.

1, 626, 392501, 246097501, 154302740626, 96747572275001, 60660573513685001, 38034082845508220626, 23847309283560140647501, 14952224886709362677762501, 9375021156657486838816440626

Offset

0,2

Comments

(25*b(n))^2 - 629*a(n)^2 = -4 with b(n)=A098261(n) give all positive solutions of this Pell equation.

Links

G. C. Greubel, Table of n, a(n) for n = 0..350

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 (627,-1).

Formula

a(n) = ((-1)^n)*S(2*n, 25*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.

G.f.: (1-x)/(1-627*x+x^2).

a(n) = S(n, 627) - S(n-1, 627) = T(2*n+1, sqrt(629)/2)/(sqrt(629)/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) = 627*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=626. - Philippe Deléham, Nov 18 2008

Example

All positive solutions of Pell equation x^2 - 629*y^2 = -4 are (25=25*1,1), (15700=25*628,626), (9843875=25*393755,392501), (6172093925=25*246883757,246097501), ...

Mathematica

LinearRecurrence[{627, -1}, {1, 626}, 20] (* G. C. Greubel, Aug 01 2019 *)

Prog

(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-627*x+x^2)) \\ G. C. Greubel, Aug 01 2019
(Magma) I:=[1, 626]; [n le 2 select I[n] else 627*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
(Sage) ((1-x)/(1-627*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 626];; for n in [3..20] do a[n]:=627*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Crossrefs

Sequence in context: A158383 A031728 A031638 * A031523 A345523 A345777

Adjacent sequences: A098259 A098260 A098261 * A098263 A098264 A098265

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 10 2004

Status

approved