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.

Index entries for sequences related to linear recurrences with constant coefficients

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.

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).

lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 10]], {n, 0, 8^2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 11 2008]

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

Cf. A001084.

Adjacent sequences: A075840 A075841 A075842 this_sequence A075844 A075845 A075846

Sequence in context: A084329 A097832 A063815 this_sequence A090051 A089957 A019319

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