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A005319
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a(n) = 6a(n-1) - a(n-2).
(Formerly M3599)
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12
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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 [From Ctibor O. Zizka, Nov 09 2009]
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REFERENCES
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John M. Campbell, An Integral Representation of Kekule' Numbers, and Double Integrals Related to Smarandache Sequences, Arxiv preprint arXiv:1105.3399, 2011.
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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Table of n, a(n) for n=0..21.
Tanya Khovanova, Recursive Sequences
_Simon 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.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
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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 (creighton.k.dement(AT)uni-oldenburg.de), Nov 17 2004
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 Jul 22 2005
a(n)=4*A001109(n). [M. Hasler, Mar 2009] [From R. J. Mathar, Jun 03 2009]
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MAPLE
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A005319:=4*z/(1-6*z+z**2); [Conjectured by Simon Plouffe in his 1992 dissertation.]
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CROSSREFS
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Sequence in context: A122690 A183512 A204199 * A155119 A114169 A121102
Adjacent sequences: A005316 A005317 A005318 * A005320 A005321 A005322
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KEYWORD
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nonn
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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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