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Search: id:A075843
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| A075843 |
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99*a(n)^2 + 1 is a square. |
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+0 5
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| 0, 1, 20, 399, 7960, 158801, 3168060, 63202399, 1260879920, 25154396001, 501827040100, 10011386405999, 199725901079880, 3984506635191601, 79490406802752140, 1585823629419851199, 31636982181594271840
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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).
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REFERENCES
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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.
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LINKS
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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
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FORMULA
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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).
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PROGRAM
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sage: [lucas_number1(n, 20, 1) for n in xrange(0, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
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CROSSREFS
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Cf. A001084.
Sequence in context: A084329 A097832 A063815 this_sequence A090051 A089957 A019319
Adjacent sequences: A075840 A075841 A075842 this_sequence A075844 A075845 A075846
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KEYWORD
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nonn
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AUTHOR
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Gregory V. Richardson (omomom(AT)hotmail.com), Oct 14 2002
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EXTENSIONS
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Chebyshev and Pell comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 2002
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