0,2

a(n) gives the general (positive integer) solution of the Pell equation b^2 - 357*a^2 =+4 with companion sequence b(n)=A078369(n+1), n>=0.

This is the m=21 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..20 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362, A007655, A078364, A077412, A078366 and A049660. 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=19, 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=21.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Zerinvary Lajos, Sage Notebooks

a(n)=19*a(n-1)-a(n-2), n >= 1; a(-1)=0, a(0)=1.

a(n)=(ap^(n+1)-am^(n+1))/(ap-am) with ap := (19+sqrt(357))/2 and am := (19-sqrt(357))/2.

a(n)=S(2*n+1, sqrt(21))/sqrt(21) = S(n, 19); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.

G.f.: 1/(1-19*x+x^2).

sage: [lucas_number1(n, 19, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008

a(n)=sqrt((A078369(n+1)^2 - 4)/357), n>=0, (Pell equation d=357, +4).

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

Sequence in context: A142549 A049629 A049664 this_sequence A045609 A128360 A001029

Adjacent sequences: A078365 A078366 A078367 this_sequence A078369 A078370 A078371

nonn,easy

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