Search: id:A005667
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%I A005667 M3056
%S A005667 1,3,19,117,721,4443,27379,168717,1039681,6406803,39480499,243289797,
%T A005667 1499219281,9238605483,56930852179,350823718557,2161873163521,13322062699683,
%U A005667 82094249361619,505887558869397,3117419602578001,19210405174337403,118379850648602419
%N A005667 Numerators of continued fraction convergents to sqrt(10).
%C A005667 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).
%C A005667 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.
%D A005667 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005667 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.
%D A005667 E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 
               7 (1969), 231-242, Thm. 1, p. 233.
%H A005667 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A005667 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005667 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005667 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A005667 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A005667 a(n) = 6a(n-1) + a(n-2).
%F A005667 G.f.: (1-3*x)/(1-6*x-x^2).
%F A005667 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.
%F A005667 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
%p A005667 A005667:=(-1+3*z)/(-1+6*z+z**2); [Conjectured by S. Plouffe in his 1992 
               dissertation.]
%Y A005667 Cf. A084134, A005668.
%Y A005667 Sequence in context: A037781 A037585 A084133 this_sequence A098444 A139176 
               A126809
%Y A005667 Adjacent sequences: A005664 A005665 A005666 this_sequence A005668 A005669 
               A005670
%K A005667 nonn,cofr
%O A005667 0,2
%A A005667 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A005667 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Jan 10 2003

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