0,3

Lim. n-> Inf. a(n)/a(n-1) = 10 + 3*Sqrt(11).

Chebyshev's polynomials U(n,x) evaluated at x=10.

The a(n) give all (unsigned, integer) solutions of Pell equation b(n)^2 - 99*a(n)^2 = +1 with b(n)= A001085(n).

A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.

L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, p. 341-400.

Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, p. 139-147.

Tanya Khovanova, Recursive Sequences

J. J. O'Connor and E. F. Robertson, Pell's Equation

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to Chebyshev polynomials.

Zerinvary Lajos, Sage Notebooks

a(n) = [(10+3*Sqrt(11))^n - (10-3*Sqrt(11))^n] / (6*Sqrt(11))

a(n) = 20*a(n-1) - a(n-2), n>=1, a(0)=0, a(1)=1.

a(n) = S(n-1, 20), with S(n, x) := U(n, x/2), Chebyshev's polynomials of the second kind. S(-1, x) := 0. See A049310.

G.f.: x/(1-20*x+x^2).

a(n) = sqrt((A001085(n)^2 - 1)/99).

sage: [lucas_number1(n, 20, 1) for n in xrange(0, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008

Cf. A001084.

Sequence in context: A084329 A097832 A063815 this_sequence A090051 A089957 A019319

Adjacent sequences: A075840 A075841 A075842 this_sequence A075844 A075845 A075846

nonn

Gregory V. Richardson (omomom(AT)hotmail.com), Oct 14 2002

Chebyshev and Pell comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 2002