A099397 Chebyshev polynomial of the first kind, T(n,x), evaluated at x=51.

1, 51, 5201, 530451, 54100801, 5517751251, 562756526801, 57395647982451, 5853793337683201, 597029524795704051, 60891157735824130001, 6210301059529265556051, 633389816914249262587201

Offset: 0

Author

Wolfdieter Lang, Oct 18 2004

Keywords

• nonn
• easy

Comments

Used in A099374.

Numbers n such that 26*(n^2-1) is square. - Vincenzo Librandi, Nov 17 2010

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..497
• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (102, -1).

Crossrefs

Row 5 of array A188645.

Programs

Magma

[n: n in [1..1000] |IsSquare(26*(n^2-1))] // Vincenzo Librandi, Nov 17 2010

Mathematica

LinearRecurrence[{102, -1},{1, 51},13] (* Ray Chandler, Aug 11 2015 *)

PARI

a(n) = polchebyshev(n, 1, 51); \\ Michel Marcus, Jan 20 2018

Formula

a(n) = 102*a(n-1) - a(n-2), n>=1; a(-1):= 51, a(0)=1.

a(n) = T(n, 51) = (S(n, 102)-S(n-2, 102))/2 = S(n, 102)-51*S(n-1, 102) with T(n, x), resp. S(n, x), Chebyshev polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 102)=(n).

a(n) = (ap^n + am^n)/2 with ap := 51+10*sqrt(26) and am := 51-10*sqrt(26).

a(n) = Sum_{k=0..floor(n/2)} (((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*51)^(n-2*k)), n >= 1. a(0):=1.

G.f.: (1 - 51*x)/(1 - 102*x + x^2).