Search: id:A001079
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%I A001079 M4005 N1659
%S A001079 1,5,49,485,4801,47525,470449,4656965,46099201,456335045,
%T A001079 4517251249,44716177445,442644523201,4381729054565,43374646022449,
%U A001079 429364731169925,4250272665676801,42073361925598085,416483346590304049
%N A001079 a(n) = 10a(n-1) - a(n-2); a(0) = 1, a(1) = 5.
%C A001079 Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where 
               r=sqrt(6) - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 14 2004
%C A001079 Appears to give all solutions >1 to the equation : x^2=ceiling(x*r*floor(x/
               r)) where r=sqrt(6). - Benoit Cloitre, Feb 24, 2004
%C A001079 (sqrt(2)+sqrt(3))^(2*n)=a(n)+A001078(n)*sqrt(6). - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Mar 12 2008
%D A001079 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001079 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001079 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 A001079 L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, 
               reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 
               1, p. 374.
%D A001079 V. Th\'{e}bault, Les R\'{e}cr\'{e}ations Math\'{e}matiques. Gauthier-Villars, 
               Paris, 1952, p. 281.
%H A001079 T. D. Noe, <a href="b001079.txt">Table of n, a(n) for n=0..200</a>
%H A001079 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001079 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 A001079 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 A001079 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A001079 L. Euler, <a href="http://www.mathematik.uni-bielefeld.de/~sieben/euler/
               euler_2.djvu">Vollstaendige Anleitung zur Algebra, Zweiter Teil</
               a>.
%H A001079 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E029.html">
               De solutione problematum diophanteorum per numeros integros</a>, 
               par. 18
%H A001079 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A001079 For all members x of the sequence, 6*x^2 -6 is a square. Lim. n-> Inf. 
               a(n)/a(n-1) = 5 + 2*Sqrt(6). - Gregory V. Richardson (omomom(AT)hotmail.com), 
               Oct 13 2002
%F A001079 a(n) = T(n, 5) = (S(n, 10)-S(n-2, 10))/2 with S(n, x) := U(n, x/2) and 
               T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. 
               second, kind. See A053120 and A049310. S(n, 10)= A004189(n+1).
%F A001079 a(n) = sqrt(1+24*A004189(n)^2) (cf. Richardson comment).
%F A001079 a(n)a(n+3) - a(n+1)a(n+2) = 240. - R. Stephan, Jun 06 2005
%F A001079 Chebyshev's polynomials T(n,x) evaluated at x=5.
%F A001079 G.f.: (1-5*x)/(1-10*x+x^2). a(n)= ((5+2*sqrt(6))^n + (5-2*sqrt(6))^n)/
               2.
%F A001079 a(-n)=a(n).
%F A001079 a(n+1)=5*a(n)+2*(6*a(n)^2-6)^0.5 - Richard Choulet (richardchoulet(AT)yahoo.fr), 
               Sep 19 2007
%F A001079 a(n+1) = 2*A054320(n) + 3*A138288(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 12 2008
%F A001079 a(n) = Cosh[2n*ArcSinh[Sqrt[2]]] - Herbert Kociemba (kociemba(AT)t-online.de), 
               Apr 24 2008
%F A001079 a(n)=(-1)^n Cos[2n ArcSin[Sqrt[3]]] [From Artur Jasinski (grafix(AT)csl.pl), 
               Oct 29 2008]
%F A001079 A001079(n) = 142238(2n-1) = A041006(2n-1) = A041038(2n-1), for all n>
               0. [From M. F. Hasler (MHasler(AT)univ-ag.fr), Feb 14 2009]
%p A001079 A001079:=-(-1+5*z)/(1-10*z+z**2); [Conjectured by S. Plouffe in his 1992 
               dissertation.]
%t A001079 Table[(-1)^n Round[N[Cos[2 n ArcSin[Sqrt[3]]], 50]], {n, 0, 20}] [From 
               Artur Jasinski (grafix(AT)csl.pl), Oct 29 2008]
%o A001079 (PARI) {a(n)=subst(poltchebi(n), 'x, 5)} /* Michael Somos Sep 05 2006 
               */
%o A001079 (PARI) {a(n)=real((5+2*quadgen(24))^n)} /* Michael Somos Sep 05 2006 
               */
%o A001079 (PARI) {a(n)=n=abs(n); polsym(1-10*x+x^2, n)[n+1]/2} /* Michael Somos 
               Sep 05 2006 */
%Y A001079 Cf. A004189.
%Y A001079 Cf. A001078 A046173 A046172 A036353.
%Y A001079 Cf. A138281.
%Y A001079 Sequence in context: A155629 A096596 A146311 this_sequence A081474 A112241 
               A116873
%Y A001079 Adjacent sequences: A001076 A001077 A001078 this_sequence A001080 A001081 
               A001082
%K A001079 nonn
%O A001079 0,2
%A A001079 N. J. A. Sloane (njas(AT)research.att.com).
%E A001079 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Nov 08 2002

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