A098253 First differences of Chebyshev polynomials S(n,363) = A098251(n) with Diophantine property.

1, 362, 131405, 47699653, 17314842634, 6285240176489, 2281524869222873, 828187242287726410, 300629687425575463957, 109127748348241605689981, 39613072020724277289999146, 14379436015774564414664000017, 5219695660654146158245742007025, 1894735145381439280878789684550058

Offset

0,2

Comments

(19*b(n))^2 - 365*a(n)^2 = -4 with b(n)=A098252(n) give all positive solutions of this Pell equation.

Links

Indranil Ghosh, Table of n, a(n) for n = 0..389

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 (363,-1).

Formula

G.f.: (1 - x)/(1 - 363*x + x^2).

a(n) = ((-1)^n)*S(2*n, 19*i) with the imaginary unit i and the S(n, x)=U(n, x/2) Chebyshev polynomials.

a(n) = S(n, 363) - S(n-1, 363) = T(2*n+1, sqrt(365)/2)/(sqrt(365)/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) = 363*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=362. - Philippe Deléham, Nov 18 2008

Example

All positive solutions of Pell equation x^2 - 365*y^2 = -4 are (19=19*1,1), (6916=19*364,362), (2510489=19*132131,131405), (911300591=19*47963189,47699653), ...

Mathematica

LinearRecurrence[{363, -1}, {1, 362}, 20] (* G. C. Greubel, Aug 01 2019 *)

Prog

(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-363*x+x^2)) \\ G. C. Greubel, Aug 01 2019
(Magma) I:=[1, 362]; [n le 2 select I[n] else 363*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
(Sage) ((1-x)/(1-363*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 362];; for n in [3..20] do a[n]:=363*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Crossrefs

Sequence in context: A200558 A212324 A007565 * A031517 A192449 A116285

Adjacent sequences: A098250 A098251 A098252 * A098254 A098255 A098256

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 10 2004

Status

approved