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A109499
Number of closed walks of length n on the complete graph on 5 nodes from a given node.
17
1, 0, 4, 12, 52, 204, 820, 3276, 13108, 52428, 209716, 838860, 3355444, 13421772, 53687092, 214748364, 858993460, 3435973836, 13743895348, 54975581388, 219902325556, 879609302220, 3518437208884, 14073748835532
OFFSET
0,3
LINKS
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
Christopher R. Kitching, Henri Kauhanen, Jordan Abbott, Deepthi Gopal, Ricardo Bermúdez-Otero, and Tobias Galla, Estimating transmission noise on networks from stationary local order, arXiv:2405.12023 [cond-mat.stat-mech], 2024. See p. 48.
FORMULA
G.f.: (1 - 3*x)/(1 - 3*x - 4*x^2).
a(n) = (4^n + 4*(-1)^n)/5.
a(n+1) = 4*A015521(n). - Paul Curtz, Nov 01 2009
a(n) = 3*a(n-1) + 4*a(n-1). - G. C. Greubel, Dec 30 2017
a(n) = A108020((n - 1) / 2) = 'ccc...c' (n digits) in base 16, for odd n. - Georg Fischer, Mar 23 2019
E.g.f.: (exp(4*x) + 4*exp(-x))/5. - G. C. Greubel, Mar 23 2019
MATHEMATICA
CoefficientList[Series[(1-3*x)/(1-3*x-4*x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, 4}, {1, 0}, 30] (* G. C. Greubel, Dec 30 2017 *)
PROG
(Magma) [(4^n + 4*(-1)^n)/5: n in [0..30]]; // Vincenzo Librandi, Aug 12 2011
(PARI) a(n)=(4^n+4*(-1)^n)/5 \\ Charles R Greathouse IV, Oct 01 2012
(Sage) [(4^n+4*(-1)^n)/5 for n in (0..30)] # G. C. Greubel, Mar 23 2019
(GAP) a:=[1, 0];; for n in [3..30] do a[n]:=3*a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Mar 23 2019
CROSSREFS
Cf. A108020 (bisection), A109502.
Sequence in context: A149411 A149412 A259274 * A282587 A188230 A124006
KEYWORD
nonn,easy,walk
AUTHOR
Mitch Harris, Jun 30 2005
EXTENSIONS
Corrected by Franklin T. Adams-Watters, Sep 18 2006
STATUS
approved