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A005389 Number of Hamiltonian circuits on 2n times 4 rectangle.
(Formerly M4228)
1

%I M4228

%S 1,6,37,236,1517,9770,62953,405688,2614457,16849006,108584525,

%T 699780452,4509783909,29063617746,187302518353,1207084188912,

%U 7779138543857,50133202843990,323086934794997,2082156365731164,13418602439355485,86477122654688250,557307869909156153

%N Number of Hamiltonian circuits on 2n times 4 rectangle.

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

%H Vincenzo Librandi, <a href="/A005389/b005389.txt">Table of n, a(n) for n = 1..200</a>

%H 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.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H T. G. Schmalz, G. E. Hite and D. J. Klein, <a href="http://dx.doi.org/10.1088/0305-4470/17/2/029">Compact self-avoiding circuits on two-dimensional lattices</a>, J. Phys. A 17 (1984), 445-453.

%F G.f.: x*(1-2*x-x^2)/(1-8*x+10*x^2+x^4). - _Ralf Stephan_, Apr 23 2004

%p A005389:=-(-1+2*z+z**2)/(1-8*z+10*z**2+z**4); [Conjectured by _Simon Plouffe_ in his 1992 dissertation.]

%p 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

%t 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 *)

%t CoefficientList[Series[(1 - 2 x - x^2)/(1 - 8 x + 10 x^2 + x^4), {x, 0, 30}], x] (* _Vincenzo Librandi_, Mar 15 2014 *)

%Y Bisection of A006864.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, _Simon Plouffe_

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Last modified June 14 20:21 EDT 2021. Contains 345038 sequences. (Running on oeis4.)