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Revision History for A003499

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Showing entries 1-10 | older changes
a(n) = 6*a(n-1) - a(n-2), with a(0) = 2, a(1) = 6.
(history; published version)
#216 by Andrew Howroyd at Thu Apr 20 13:18:54 EDT 2023
STATUS

reviewed

approved

#215 by Amiram Eldar at Thu Apr 20 13:10:03 EDT 2023
STATUS

proposed

reviewed

#214 by Michel Marcus at Thu Apr 20 13:06:07 EDT 2023
STATUS

editing

proposed

#213 by Michel Marcus at Thu Apr 20 13:06:02 EDT 2023
LINKS

S. Falcon, <a href="http://dx.doi.org/10.4236/am.2014.515216">Relationships between Some k-Fibonacci Sequences, </a>, Applied Mathematics</a>, , 2014, 5, 2226-2234.

STATUS

approved

editing

#212 by Alois P. Heinz at Sun Jun 19 17:20:51 EDT 2022
STATUS

proposed

approved

#211 by Jon E. Schoenfield at Sun Jun 19 16:44:27 EDT 2022
STATUS

editing

proposed

#210 by Jon E. Schoenfield at Sun Jun 19 16:44:02 EDT 2022
LINKS

W. Lu and F. Y. Wu, <a href="https://arxiv.org/abs/cond-mat/0110035">Close-packed dimers on nonorientable surfaces</a>, arXiv:cond-mat/0110035 [cond-mat.stat-mech], 2001-2002; Physics Letters A, 293(2002), 235-246. [From Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), , Jul 04 2009]

STATUS

proposed

editing

#209 by Jon E. Schoenfield at Sun Jun 19 16:42:06 EDT 2022
STATUS

editing

proposed

#208 by Jon E. Schoenfield at Sun Jun 19 16:41:49 EDT 2022
COMMENTS

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. - _Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), _, Jul 04 2009

REFERENCES

Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002; ppp. 480-481.

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, 2001, Wiley, ppp. 77-79.

FORMULA

a(n) = Product_{r=1..n} (4*sin^2((4*r-1)*Pi/(4*n)) + 4). [Lu/Wu] - _Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), _, Jul 04 2009

Also F(-alpha) = 0.83251 21926 93800 07634 83251219269380007634 ... has the continued fraction representation 1 - 1/(6 - 1/(34 - 1/(198 - ...))) and the simple continued fraction expansion 1/(1 + 1/((6-2) + 1/(1 + 1/((34-2) + 1/(1 + 1/((198-2) + 1/(1 + ...))))))). Cf. A174501 and A003500.

G.f.: G(0), where G(k) = 1 + 1/(1 - x*(8*k-9)/( x*(8*k-1) - 3/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 12 2013

PROG

(MAGMAMagma) I:=[2, 6]; [n le 2 select I[n] else 6*Self(n-1) -Self(n-2): n in [1..25]]; // G. C. Greubel, Jan 16 2020

(MAGMAMagma) R<x>:=PowerSeriesRing(Integers(), 25); Coefficients(R!( (2-6*x)/(1 - 6*x + x^2) )); // Marius A. Burtea, Jan 16 2020

STATUS

approved

editing

#207 by Charles R Greathouse IV at Wed Apr 13 13:25:17 EDT 2022
LINKS

Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Discussion
Wed Apr 13
13:25
OEIS Server: https://oeis.org/edit/global/2938