Search: id:A075843
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%I A075843
%S A075843 0,1,20,399,7960,158801,3168060,63202399,1260879920,25154396001,
%T A075843 501827040100,10011386405999,199725901079880,3984506635191601,
%U A075843 79490406802752140,1585823629419851199,31636982181594271840
%N A075843 99*a(n)^2 + 1 is a square.
%C A075843 Lim. n-> Inf. a(n)/a(n-1) = 10 + 3*Sqrt(11).
%C A075843 Chebyshev's polynomials U(n,x) evaluated at x=10.
%C A075843 The a(n) give all (unsigned, integer) solutions of Pell equation b(n)^2 
               - 99*a(n)^2 = +1 with b(n)= A001085(n).
%D A075843 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.
%D A075843 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine 
               Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, 
               p. 341-400.
%D A075843 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.
%H A075843 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A075843 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A075843 J. J. O'Connor and E. F. Robertson, <a href="http://www-gap.dcs.st-and.ac.uk/
               ~history/HistTopics/Pell.html">Pell's Equation</a>
%H A075843 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PellEquation.html">Link to a section of The World of Mathematics.</
               a>
%H A075843 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A075843 a(n) = [(10+3*Sqrt(11))^n - (10-3*Sqrt(11))^n] / (6*Sqrt(11))
%F A075843 a(n) = 20*a(n-1) - a(n-2), n>=1, a(0)=0, a(1)=1.
%F A075843 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.
%F A075843 G.f.: x/(1-20*x+x^2).
%F A075843 a(n) = sqrt((A001085(n)^2 - 1)/99).
%t A075843 lst={};Do[AppendTo[lst, GegenbauerC[n, 1, 10]], {n, 0, 8^2}];lst [From 
               Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 11 2008]
%o A075843 sage: [lucas_number1(n,20,1) for n in xrange(0,20)] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 25 2008
%Y A075843 Cf. A001084.
%Y A075843 Sequence in context: A097832 A063815 A158534 this_sequence A090051 A089957 
               A019319
%Y A075843 Adjacent sequences: A075840 A075841 A075842 this_sequence A075844 A075845 
               A075846
%K A075843 nonn
%O A075843 0,3
%A A075843 Gregory V. Richardson (omomom(AT)hotmail.com), Oct 14 2002
%E A075843 Chebyshev and Pell comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Nov 08 2002

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