A109499 Number of closed walks of length n on the complete graph on 5 nodes from a given node.

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

Author

Mitch Harris, Jun 30 2005

Keywords

• nonn
• easy
• walk

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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.
• Index entries for linear recurrences with constant coefficients, signature (3,4).

Crossrefs

Cf. A108020 (bisection), A109502.

Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521 (forms k=4^n-k). - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Programs

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

Magma

[(4^n + 4*(-1)^n)/5: n in [0..30]]; // Vincenzo Librandi, Aug 12 2011

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 *)

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

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

Extensions

Corrected by Franklin T. Adams-Watters, Sep 18 2006