0,2

a(2*n+1) with b(2*n+1) := A041024(2*n+1), n>=0, give all (positive integer) solutions to Pell equation b^2 - 17*a^2 = +1, a(2*n) with b(2*n) := A041024(2*n), n>=0, give all (positive integer) solutions to Pell equation b^2 - 17*a^2 = -1 (cf. Emerson reference).

Bisection: a(2*n)= T(2*n+1,sqrt(17))/sqrt(17)= A078988(n), n>=0, and a(2*n+1)=8*S(n-1,66),n>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first,resp. second kind. S(-1,x)=0. See A053120, resp. A049310.

Sqrt(17) = 8/2 + 8/65 + 8/(65*4289) + 8/(4289*283009)... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 26 2007

E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Thm. 1, p. 233.

S. Falcon & A. Plaza: The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals (2007)

S. Falcon & A. Plaza: On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals (2007)

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

G.f.: 1/(1-8*x-x^2).

a(n) = ((-i)^n)*S(n, 8*i), with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind, and i^2=-1. See A049310.

a(n)=F(n, 8), the n-th Fibonacci polynomial evaluated at x=8. - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006

a(n) = ((4+Sqrt[17])^n-(4-Sqrt[17])^n)/(2Sqrt{17]); a(n) = Sum[Binomial[n-1-i,i]*8^{n-1-2i}, {i,0,Floor[(n-1)/2]}] - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Sep 24 2007

Let T = the 2 X 2 matrix [0, 1; 1, 8]. Then T^n * [1, 0] = [a(n-2), a(n-1)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 26 2007

Cf. A041024.

Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413.

Sequence in context: A033118 A033126 A022039 this_sequence A081190 A024105 A041114

Adjacent sequences: A041022 A041023 A041024 this_sequence A041026 A041027 A041028

nonn,cofr,easy

njas

Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 10 2003