Search: id:A003501
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%I A003501 M1540
%S A003501 2,5,23,110,527,2525,12098,57965,277727,1330670,6375623,30547445,
%T A003501 146361602,701260565,3359941223,16098445550,77132286527,369562987085,
%U A003501 1770682648898,8483850257405,40648568638127,194758992933230
%N A003501 a(n) = 5a(n-1) - a(n-2).
%C A003501 Positive values of x satisfying x^2 - 21*y^2 = 4; values of y are in 
               A004254. - W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Nov 29 2002
%D A003501 J. O. Shallit, An interesting continued fraction, Math. Mag., 48 (1975), 
               207-211.
%D A003501 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A003501 T. D. Noe, <a href="b003501.txt">Table of n, a(n) for n=0..200</a>
%H A003501 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A003501 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 A003501 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 A003501 <a href="Sindx_Rea.html#recur1">Index entries for recurrences a(n) = 
               k*a(n - 1) +/- a(n - 2)</a>
%H A003501 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%H A003501 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A003501 a(n) = 5*S(n-1, 5) - 2*S(n-2, 5) = S(n, 5) - S(n-2, 5) = 2*T(n, 5/2), 
               with S(n, x)=U(n, x/2), S(-1, x)=0, S(-2, x)=-1. U(n, x), resp. T(n, 
               x), are Chebyshev's polynomials of the second, resp. first, kind. 
               S(n-1, 5) = A004254(n), n>=0.
%F A003501 G.f.: (2-5*x)/(1-5*x+x^2).
%F A003501 a(n) ~ (1/2*(5 + sqrt(21)))^n - Joe Keane (jgk(AT)jgk.org), May 16 2002
%F A003501 a(n) = ap^n + am^n, with ap=(5+sqrt(21))/2 and am=(5-sqrt(21))/2.
%p A003501 A003501:=-(-2+5*z)/(1-5*z+z**2); [Conjectured by S. Plouffe in his 1992 
               dissertation.]
%t A003501 a[0] = 2; a[1] = 5; a[n_] := 5a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 
               21}] (from Robert G. Wilson v Jan 30 2004)
%o A003501 (PARI) a(n)=if(n<0,0,subst(poltchebi(n),x,5/2)*2)
%o A003501 sage: [lucas_number2(n,5,1) for n in range(37)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 25 2008
%Y A003501 Cf. A004252, A004253. a(n) = sqrt(4+21*A004254(n)^2).
%Y A003501 Sequence in context: A106858 A100299 A038833 this_sequence A006990 A032182 
               A165902
%Y A003501 Adjacent sequences: A003498 A003499 A003500 this_sequence A003502 A003503 
               A003504
%K A003501 nonn
%O A003501 0,1
%A A003501 N. J. A. Sloane (njas(AT)research.att.com).
%E A003501 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 31 2000
%E A003501 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Oct 31 2002

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