0,2

a(2*n+1) with b(2*n+1) := A005668(2*n+1), n>=0, give all (positive integer) solutions to Pell equation a^2 - 10*b^2 = -1, a(2*n) with b(2*n) := A005668(2*n), n>=1, give all (positive integer) solutions to Pell equation a^2 - 10*b^2 = +1 (cf. Emerson reference).

Bisection: a(2*n)= T(n,19)=A078986(n), n>=0, and a(2*n+1)=3*S(2*n,2*sqrt(10)),n>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first,resp. second kind. See A053120, resp. A049310.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

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

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

a(n) = 6a(n-1) + a(n-2).

G.f.: (1-3*x)/(1-6*x-x^2).

a(n) = ((-i)^n)*T(n, 3*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120), and i^2=-1.

Binomial transform of A084132. E.g.f. : exp(3x)cosh(sqrt(10)x); a(n)=((3+sqrt(10))^n+(3-sqrt(10))^n)/2; a(n)=sum{k=0..floor(n/2), C(n, 2k)10^k3^(n-2k)}. - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003

A005667:=(-1+3*z)/(-1+6*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

Cf. A084134, A005668.

Adjacent sequences: A005664 A005665 A005666 this_sequence A005668 A005669 A005670

Sequence in context: A037781 A037585 A084133 this_sequence A098444 A139176 A126809

nonn,cofr

njas, R. K. Guy

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