%I A005319 M3599
%S A005319 0,4,24,140,816,4756,27720,161564,941664,5488420,31988856,186444716,
%T A005319 1086679440,6333631924,36915112104,215157040700,1254027132096,7309005751876,
%U A005319 42600007379160,248291038523084,1447146223759344,8434586304032980
%N A005319 a(n) = 6a(n-1) - a(n-2).
%C A005319 Solutions y of the equation 2x^2-y^2=2; the corresponding x values are 
               given by A001541. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Feb 
               25 2008
%C A005319 The lower intermediate convergents to 2^(1/2) beginning with 4/3, 24/
               17, 140/99, 816/577, form a strictly increasing sequence; essentially, 
               numerators=A005319 and denominators=A001541. - Clark Kimberling (ck6(AT)evansville.edu), 
               Aug 26 2008
%C A005319 Numbers n such that (ceiling(sqrt(n*n/2)))^2 = 1 + n*n/2 [From Ctibor 
               O. Zizka (c.zizka(AT)email.cz), Nov 09 2009]
%D A005319 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005319 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 A005319 P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 
               2000, p. 160, middle display.
%H A005319 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A005319 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 A005319 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 A005319 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%F A005319 G.f. for signed version beginning with 1: (1+2*x+x^2)/(1+6*x+x^2).
%F A005319 For any term n of the sequence, 2*n^2 + 4 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 A005319 a(n) = [(3+2*Sqrt(2))^n - (3-2*Sqrt(2))^n] / Sqrt(2) - Gregory V. Richardson 
               (omomom(AT)hotmail.com), Oct 06 2002
%F A005319 (-1)^(n+1) = A090390(n+1) + A001542(n+1) + A046729(n) - a(n) (conjectured). 
               Generated by the floretion - .5'i + .5'j - .5i' + .5j' + 'ii' - 'jj' 
               - 2'kk' + 'ij' + .5'ik' + 'ji' + .5'jk' + .5'ki' + .5'kj' + e - Creighton 
               Dement (creighton.k.dement(AT)uni-oldenburg.de), Nov 17 2004
%F A005319 For n>0, a(n)=A000129(n+1)^2-A000129(n-1)^2; e.g. 816=29^2-5^2; a(n)=A046090(n-1)+A001652(n); 
               e.g. 816=120+696; a(n)=A001653(n)-A001653(n-1); e.g. 816=985-169 
               - Charlie Marion (charliemath(AT)optonline.net) Jul 22 2005
%F A005319 a(n)=4*A001109(n). [M. Hasler, Mar 2009] [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Jun 03 2009]
%p A005319 A005319:=4*z/(1-6*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
%Y A005319 Sequence in context: A071079 A153337 A122690 this_sequence A155119 A114169 
               A121102
%Y A005319 Adjacent sequences: A005316 A005317 A005318 this_sequence A005320 A005321 
               A005322
%K A005319 nonn,new
%O A005319 0,2
%A A005319 N. J. A. Sloane (njas(AT)research.att.com).