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A099370 Chebyshev's polynomial of the first kind, T(n,x), evaluated at x=33. 5
1, 33, 2177, 143649, 9478657, 625447713, 41270070401, 2723199198753, 179689877047297, 11856808685922849, 782369683393860737, 51624542295308885793, 3406437421806992601601, 224773245296966202819873 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Used in A099369.

Solutions of the Pell equation x^2 - 17y^2 = 1 (x values). After initial term this sequence bisects A041024. See 8*A097316(n-1) with A097316(-1) = 0 for corresponding y values. a(n+1)/a(n) apparently converges to (4+sqrt(17))^2. (See related comments in A088317, which this sequence also bisects.). - Rick L. Shepherd, Jul 31 2006

From a(n) = T(n, 33) (see the formula section) and the de Moivre-Binet formula for T(n,x=33) follows a(n+1)/a(n) = 33  + 8*sqrt(17), which is the conjectured value (4+sqrt(17))^2 given in the previous comment by Rick L. Shepherd. - Wolfdieter Lang, Jun 28 2013

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Tanya Khovanova, Recursive Sequences

Eric Weisstein's World of Mathematics, Pell Equation

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (66, -1).

FORMULA

a(n)= 66*a(n-1) - a(n-2), a(-1):= 33, a(0)=1.

a(n)= T(n, 33)= (S(n, 66)-S(n-2, 66))/2 = S(n, 66)-33*S(n-1, 66) with T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp.second, kind. See A053120 and A049310. S(n, 66)=A097316(n).

a(n)= (ap^n + am^n)/2 with ap := 33+8*sqrt(17) and am := 33-8*sqrt(17).

a(n)= sum(((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*33)^(n-2*k), k=0..floor(n/2)), n>=1. a(0):=1.

G.f.: (1-33*x)/(1-66*x+x^2).

EXAMPLE

a(1)^2 - 17*A121470(1)^2 = 33^2 - 17*8^2 = 1089 - 1088 = 1.

MATHEMATICA

LinearRecurrence[{66, -1}, {1, 33}, 14] (* Ray Chandler, Aug 11 2015 *)

PROG

(PARI) Program uses fact that continued fraction for sqrt(17) = [4, 8, 8, ...]. print1("1, "); forstep(n=2, 40, 2, v=vector(n, i, if(i>1, 8, 4)); print1(contfracpnqn(v)[1, 1], ", ")) - Rick L. Shepherd, Jul 31 2006

CROSSREFS

Cf. A121470, A041024, A040012.

Row 4 of array A188645.

Sequence in context: A180920 A120288 A284112 * A267672 A283533 A118641

Adjacent sequences:  A099367 A099368 A099369 * A099371 A099372 A099373

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Oct 18 2004

EXTENSIONS

A-number for y values in Pell equation corrected. - Wolfdieter Lang, Jun 28 2013

STATUS

approved

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Last modified June 22 07:47 EDT 2017. Contains 288605 sequences.