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A005319
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a(n) = 6*a(n-1) - a(n-2).
(Formerly M3599)
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17
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0, 4, 24, 140, 816, 4756, 27720, 161564, 941664, 5488420, 31988856, 186444716, 1086679440, 6333631924, 36915112104, 215157040700, 1254027132096, 7309005751876, 42600007379160, 248291038523084, 1447146223759344, 8434586304032980
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OFFSET
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0,2
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COMMENTS
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Solutions y of the equation 2x^2-y^2=2; the corresponding x values are given by A001541. - N-E. Fahssi, Feb 25 2008
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
Numbers n such that (ceiling(sqrt(n*n/2)))^2 = 1 + n*n/2. - Ctibor O. Zizka, Nov 09 2009
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
Also the number of edge covers in the n-triangular snake graph. - Eric W. Weisstein, Jun 08 2019
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REFERENCES
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P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 160, middle display.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Marius A. Burtea, Table of n, a(n) for n = 0..100
Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
Hacène Belbachir, Soumeya Merwa Tebtoub, László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Soumeya M. Tebtoub, Hacène Belbachir, László Németh, Integer sequences and ellipse chains inside a hyperbola, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.
Eric Weisstein's World of Mathematics, Edge Cover
Eric Weisstein's World of Mathematics, Triangular Snake Graph
Index entries for linear recurrences with constant coefficients, signature (6,-1).
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FORMULA
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G.f.: 4*x / ( 1-6*x+x^2 ). - Simon Plouffe in his 1992 dissertation.
G.f. for signed version beginning with 1: (1+2*x+x^2)/(1+6*x+x^2).
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
a(n) = ((3+2*sqrt(2))^n - (3-2*sqrt(2))^n) / sqrt(2). - Gregory V. Richardson, Oct 06 2002
(-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
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
a(n) = 4*A001109(n). - M. F. Hasler, Mar 2009
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
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MATHEMATICA
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LinearRecurrence[{6, -1}, {0, 4}, 22] (* Jean-François Alcover, Sep 26 2017 *)
Table[((3 + 2 Sqrt[2])^n - (3 - 2 Sqrt[2])^n)/Sqrt[2], {n, 20}] // Expand (* Eric W. Weisstein, Jun 08 2019 *)
CoefficientList[Series[(4 x)/(1 - 6 x + x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 08 2019 *)
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PROG
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(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
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CROSSREFS
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Cf. A000129, A001109, A001541, A001542, A001652, A001653, A046090, A046729, A090390
Sequence in context: A183512 A204199 A262376 * A155119 A114169 A121102
Adjacent sequences: A005316 A005317 A005318 * A005320 A005321 A005322
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KEYWORD
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nonn,easy,changed
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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