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A005667
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Numerators of continued fraction convergents to sqrt(10).
(Formerly M3056)
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8
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1, 3, 19, 117, 721, 4443, 27379, 168717, 1039681, 6406803, 39480499, 243289797, 1499219281, 9238605483, 56930852179, 350823718557, 2161873163521, 13322062699683, 82094249361619, 505887558869397, 3117419602578001, 19210405174337403, 118379850648602419
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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.
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REFERENCES
| E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Thm. 1, p. 233.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Tanya Khovanova, Recursive Sequences
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.
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| 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
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MAPLE
| A005667:=(-1+3*z)/(-1+6*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Join[{1}, Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[10], n]]], {n, 1, 50}]] (*From Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)
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CROSSREFS
| Cf. A084134, A005668.
Sequence in context: A037781 A037585 A084133 * A098444 A139176 A126809
Adjacent sequences: A005664 A005665 A005666 * A005668 A005669 A005670
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KEYWORD
| nonn,frac
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
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EXTENSIONS
| Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 10 2003
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