0,2

a(n) gives the general (positive integer) solution of the Pell equation b^2 - 285*a^2 =+4 with companion sequence b(n)=A078367(n+1), n>=0.

This is the m=19 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..18 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362, A007655, A078364 and A077412. The m=1..3 (signed) sequences are A049347, A056594, A010892.

A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=17, q=-1.

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=19.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Zerinvary Lajos, Sage Notebooks

a(n)=17*a(n-1)-a(n-2), n >= 1; a(-1)=0, a(0)=1.

a(n)=S(2*n+1, sqrt(19))/sqrt(19) = S(n, 17); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.

a(n)=(ap^(n+1)-am^(n+1))/(ap-am) with ap := (17+sqrt(285))/2 and am := (17-sqrt(285))/2.

G.f.: 1/(1-17*x+x^2).

sage: [lucas_number1(n, 17, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008

a(n)=sqrt((A078367(n+1)^2 - 4)/285), n>=0, (Pell equation d=285, +4).

Cf. A077428, A078355 (Pell +4 equations).

Sequence in context: A029811 A113076 A012235 this_sequence A045607 A045606 A128358

Adjacent sequences: A078363 A078364 A078365 this_sequence A078367 A078368 A078369

nonn,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002