0,2

Chebyshev or generalized Fibonacci sequence.

This is the m=13 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..12 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, and A004189. The m=1..3 (signed) sequences are A049347, A056594, A010892.

All positive integer solutions of Pell equation b(n)^2 - 117*a(n)^2 = +4 together with b(n+1)=A057076(n+1), n>=0. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 31 2004

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=11, 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=13.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Zerinvary Lajos, Sage Notebooks

Recursion: a(n)=11*a(n-1)-a(n-2), n >= 1; a(-1)=0, a(0)=1.

a(n)=S(2*n+1, sqrt(13))/sqrt(13) = S(n, 11); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.

G.f.: 1/(1-11*x+x^2).

with (combinat):seq(fibonacci(2*n, 3)/3, n=1..18); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 20 2008

sage: [lucas_number1(n, 11, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008

A049310, A004189. a(n)=sqrt((A057076(n+1)^2 - 4)/117).

Sequence in context: A060499 A060498 A081122 this_sequence A089707 A084969 A045592

Adjacent sequences: A004187 A004188 A004189 this_sequence A004191 A004192 A004193

nonn

njas

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 31 2002