A005389 Number of Hamiltonian circuits on 2n times 4 rectangle. (Formerly M4228)

1, 6, 37, 236, 1517, 9770, 62953, 405688, 2614457, 16849006, 108584525, 699780452, 4509783909, 29063617746, 187302518353, 1207084188912, 7779138543857, 50133202843990, 323086934794997, 2082156365731164, 13418602439355485, 86477122654688250, 557307869909156153

Offset

1,2

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

T. G. Schmalz, G. E. Hite and D. J. Klein, Compact self-avoiding circuits on two-dimensional lattices, J. Phys. A 17 (1984), 445-453.

Index entries for linear recurrences with constant coefficients, signature (8,-10,0,-1).

Formula

G.f.: x*(1-2*x-x^2)/(1-8*x+10*x^2+x^4). - Ralf Stephan, Apr 23 2004

Maple

A005389:=-(-1+2*z+z**2)/(1-8*z+10*z**2+z**4); [Conjectured by Simon Plouffe in his 1992 dissertation.]
a:= n -> (Matrix([[0, 1, 2, -11]]). Matrix(4, (i, j)-> if (i=j-1) then 1 elif j=1 then [8, -10, 0, -1][i] else 0 fi)^(n))[1, 1]: seq (a(n), n=1..25); # Alois P. Heinz, Aug 05 2008

Mathematica

a[1]=1; a[2]=6; a[3]=37; a[4]=236; a[n_] := a[n] = 8*a[n-1]-10*a[n-2]-a[n-4]; Array[a, 23] (* Jean-François Alcover, Mar 13 2014 *)
CoefficientList[Series[(1 - 2 x - x^2)/(1 - 8 x + 10 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 15 2014 *)

Prog

(Magma) I:=[1, 6, 37, 236]; [n le 4 select I[n] else 8*Self(n-1) -10*Self(n-2) -Self(n-4): n in [1..41]]; // G. C. Greubel, Nov 17 2022
(SageMath)
def A005389_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x-x^2)/(1-8*x+10*x^2+x^4) ).list()
A005389_list(40) # G. C. Greubel, Nov 17 2022

Crossrefs

Bisection of A006864.

Sequence in context: A218186 A154623 A196834 * A080954 A271905 A351152

Adjacent sequences: A005386 A005387 A005388 * A005390 A005391 A005392

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Status

approved