Search: id:A003499
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%I A003499 M1701
%S A003499 2,6,34,198,1154,6726,39202,228486,1331714,7761798,45239074,263672646,
%T A003499 1536796802,8957108166,52205852194,304278004998,1773462177794,
%U A003499 10336495061766,60245508192802,351136554095046,2046573816377474
%N A003499 a(0) = 2, a(1) = 6; for n >= 2, a(n) = 6a(n-1) - a(n-2).
%C A003499 Two times Chebyshev polynomials of the first kind evaluated at 3.
%C A003499 Also 2(a(2n)-2) and a(2n+1)-2 are perfect squares. - Mario Catalani (mario.catalani(AT)unito.it), 
               Mar 31 2003
%C A003499 Chebyshev polynomials of the first kind evaluated at 3, then multiplied 
               by 2. - Michael Somos, Apr 07 2003
%C A003499 Also gives solutions >2 to the equation x^2-3 = floor(x*r*floor(x/r)) 
               where r=sqrt(2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 
               14 2004
%C A003499 Output of Lu and Wu's formula for the number of perfect matchings of 
               an m x n Klein bottle where m and n are both even specializes to 
               this sequence for m=2. [From Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), 
               Jul 04 2009]
%D A003499 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003499 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 A003499 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 198.
%D A003499 J. O. Shallit, An interesting continued fraction, Math. Mag., 48 (1975), 
               207-211.
%D A003499 Thomas Koshy, Fibonacci and Lucas Numbers with Applications, 2001, Wiley, 
               p. 77-79.
%D A003499 Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and 
               Number, World Scientific, 2002; p. 480-481.
%D A003499 W. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Physics 
               Letters A, 293(2002), 235-246. [From Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), 
               Jul 04 2009]
%H A003499 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A003499 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A003499 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 A003499 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 A003499 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A003499 <a href="Sindx_Rea.html#recur1">Index entries for recurrences a(n) = 
               k*a(n - 1) +/- a(n - 2)</a>
%F A003499 G.f.: (2-6x)/(1-6x+x^2). a(n)=(3+2*sqrt(2))^n+(3-2*sqrt(2))^n.
%F A003499 For all sequence elements n, 2*n^2 - 8 is a perfect square. Lim a(n)/
               a(n-1) = 3 + 2*sqrt(2) - Gregory V. Richardson (omomom(AT)hotmail.com), 
               Oct 06 2002
%F A003499 a(2n)+2 is a perfect square, 2(a(2n+1)+2) is a perfect square. a(n), 
               a(n-1) and A077445(n), n>0, satisfy the Diophantine equation x^2+y^2-3z^2=-8. 
               - Mario Catalani (mario.catalani(AT)unito.it), Mar 24 2003
%F A003499 a(n+1)=Trace of n-th power of matrix {{6, -1}, {1, 0}} - Artur Jasinski 
               (grafix(AT)csl.pl), Apr 22 2008
%F A003499 \prod_{r=1}^{n}(4\sin^2((4r-1)\pi/(4n))+4) (Lu/Wu) [From Sarah-Marie 
               Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009]
%p A003499 A003499:=-2*(-1+3*z)/(1-6*z+z**2); [Conjectured by S. Plouffe in his 
               1992 dissertation.]
%t A003499 a[0] = 2; a[1] = 6; a[n_] := 6a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 
               19}] (from Robert G. Wilson v Jan 30 2004)
%t A003499 Table[Tr[MatrixPower[{{6, -1}, {1, 0}}, n]], {n, 1, 100}] - Artur Jasinski 
               (grafix(AT)csl.pl), Apr 22 2008
%o A003499 (PARI) a(n)=2*real((3+quadgen(32))^n)
%o A003499 (PARI) a(n)=2*subst(poltchebi(abs(n)),x,3)
%o A003499 (PARI) a(n)=if(n<0,a(-n),polsym(1-6*x+x^2,n)[n+1])
%o A003499 sage: [lucas_number2(n,6,1) for n in range(37)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 25 2008
%Y A003499 a(n)=2 A001541(n). A081555(n)=1+a(n).
%Y A003499 Bisection of A002203.
%Y A003499 First row of array A103999.
%Y A003499 Sequence in context: A026976 A026951 A030233 this_sequence A018953 A009199 
               A052824
%Y A003499 Adjacent sequences: A003496 A003497 A003498 this_sequence A003500 A003501 
               A003502
%K A003499 nonn
%O A003499 0,1
%A A003499 N. J. A. Sloane (njas(AT)research.att.com).

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