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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

Index entries for sequences related to linear recurrences with constant coefficients

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.

Sequence in context: A037781 A037585 A084133 this_sequence A098444 A139176 A126809

Adjacent sequences: A005664 A005665 A005666 this_sequence A005668 A005669 A005670

nonn,cofr

N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy

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