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A005319 a(n) = 6*a(n-1) - a(n-2).
(Formerly M3599)
17

%I M3599

%S 0,4,24,140,816,4756,27720,161564,941664,5488420,31988856,186444716,

%T 1086679440,6333631924,36915112104,215157040700,1254027132096,

%U 7309005751876,42600007379160,248291038523084,1447146223759344,8434586304032980

%N a(n) = 6*a(n-1) - a(n-2).

%C Solutions y of the equation 2x^2-y^2=2; the corresponding x values are given by A001541. - _N-E. Fahssi_, Feb 25 2008

%C 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_, Aug 26 2008

%C Numbers n such that (ceiling(sqrt(n*n/2)))^2 = 1 + n*n/2. - _Ctibor O. Zizka_, Nov 09 2009

%C All nonnegative solutions of the indefinite binary quadratic form X^2 + 4*X*Y -4*Y^2 of discriminant 32, representing -4 are (X(n), Y(n)) = (a(n), A001653(n+1)), for n >= 0. - _Wolfdieter Lang_, Jun 13 2018

%C Also the number of edge covers in the n-triangular snake graph. - _Eric W. Weisstein_, Jun 08 2019

%D P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 160, middle display.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Marius A. Burtea, <a href="/A005319/b005319.txt">Table of n, a(n) for n = 0..100</a>

%H Andersen, K., Carbone, L. and Penta, D., <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.

%H John M. Campbell, <a href="http://arxiv.org/abs/1105.3399">An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences</a>, arXiv preprint arXiv:1105.3399 [math.GM], 2011.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EdgeCover.html">Edge Cover</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularSnakeGraph.html">Triangular Snake Graph</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).

%F G.f.: 4*x / ( 1-6*x+x^2 ). - _Simon Plouffe_ in his 1992 dissertation.

%F G.f. for signed version beginning with 1: (1+2*x+x^2)/(1+6*x+x^2).

%F 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_, Oct 06 2002

%F a(n) = ((3+2*sqrt(2))^n - (3-2*sqrt(2))^n) / sqrt(2). - _Gregory V. Richardson_, Oct 06 2002

%F (-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_, Nov 17 2004

%F For n>0, a(n)=A000129(n+1)^2-A000129(n-1)^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_ Jul 22 2005

%F a(n) = 4*A001109(n). - _M. F. Hasler_, Mar 2009

%F For n>1, a(n) is the denominator of continued fraction [1,4,1,4,...,1,4] with (n-1) repetitions of 1,4. For the numerators, see A001653. - _Greg Dresden_, Sep 10 2019

%t LinearRecurrence[{6, -1}, {0, 4}, 22] (* _Jean-François Alcover_, Sep 26 2017 *)

%t Table[((3 + 2 Sqrt[2])^n - (3 - 2 Sqrt[2])^n)/Sqrt[2], {n, 20}] // Expand (* _Eric W. Weisstein_, Jun 08 2019 *)

%t CoefficientList[Series[(4 x)/(1 - 6 x + x^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Jun 08 2019 *)

%o (MAGMA) a:=[0,4]; [n le 2 select a[n] else 6*Self(n-1) - Self(n-2):n in [1..22]]; // _Marius A. Burtea_, Sep 19 2019

%Y Cf. A000129, A001109, A001541, A001542, A001652, A001653, A005319, A046090, A046729, A090390

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

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Last modified November 19 20:42 EST 2019. Contains 329323 sequences. (Running on oeis4.)