A098256 First differences of Chebyshev polynomials S(n,443)=A098254(n) with Diophantine property.

1, 442, 195805, 86741173, 38426143834, 17022694977289, 7541015448795193, 3340652821121293210, 1479901658741284096837, 655593094169567733605581, 290426260815459764703175546, 128658177948154506195773161297, 56995282404771630784962807279025, 25248781447135884283232327851446778

Offset

0,2

Comments

(21*b(n))^2 - 445*a(n)^2 = -4 with b(n)=A098255(n) give all positive solutions of this Pell equation.

Links

Indranil Ghosh, Table of n, a(n) for n = 0..377

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 (443, -1).

Formula

G.f.: (1 - x)/(1 - 443*x + x^2).

a(n) = ((-1)^n)*S(2*n, 21*i) with the imaginary unit i and the S(n, x)=U(n, x/2) Chebyshev polynomials.

a(n) = S(n, 443) - S(n-1, 443) = T(2*n+1, sqrt(445)/2)/(sqrt(445)/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) = 443*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=442. - Philippe Deléham, Nov 18 2008

Example

All positive solutions of Pell equation x^2 - 445*y^2 = -4 are (21=21*1,1), (9324=21*444,442), (4130511=21*196691,195805), (1829807049=21*87133669,86741173), ...

Mathematica

LinearRecurrence[{443, -1}, {1, 442}, 20] (* G. C. Greubel, Aug 01 2019 *)

Prog

(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-443*x+x^2)) \\ G. C. Greubel, Aug 01 2019
(Magma) I:=[1, 442]; [n le 2 select I[n] else 443*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
(Sage) ((1-x)/(1-443*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 442];; for n in [3..20] do a[n]:=443*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Crossrefs

Sequence in context: A250086 A018237 A264270 * A225350 A171392 A105980

Adjacent sequences: A098253 A098254 A098255 * A098257 A098258 A098259

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 10 2004

Status

approved