0,2

Used in A099369.

Solutions of the Pell equation x^2 - 17y^2 = 1 (x values). After initial term this sequence bisects A041024. See A121470 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 (rshepherd2(AT)hotmail.com), Jul 31 2006

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Eric Weisstein's World of Mathematics, Pell Equation

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).

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

(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 (rshepherd2(AT)hotmail.com), Jul 31 2006

Cf. A121470, A041024, A040012.

Sequence in context: A012805 A093756 A120288 this_sequence A118641 A111922 A136541

Adjacent sequences: A099367 A099368 A099369 this_sequence A099371 A099372 A099373

nonn,easy

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 18 2004