This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A005389 M4228 #51 Nov 18 2022 03:40:59
%S A005389 1,6,37,236,1517,9770,62953,405688,2614457,16849006,108584525,
%T A005389 699780452,4509783909,29063617746,187302518353,1207084188912,
%U A005389 7779138543857,50133202843990,323086934794997,2082156365731164,13418602439355485,86477122654688250,557307869909156153
%N A005389 Number of Hamiltonian circuits on 2n times 4 rectangle.
%D A005389 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005389 Vincenzo Librandi, <a href="/A005389/b005389.txt">Table of n, a(n) for n = 1..200</a>
%H A005389 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.
%H A005389 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A005389 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.
%H A005389 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-10,0,-1).
%F A005389 G.f.: x*(1-2*x-x^2)/(1-8*x+10*x^2+x^4). - _Ralf Stephan_, Apr 23 2004
%p A005389 A005389:=-(-1+2*z+z**2)/(1-8*z+10*z**2+z**4); [Conjectured by _Simon Plouffe_ in his 1992 dissertation.]
%p A005389 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 A005389 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 A005389 CoefficientList[Series[(1 - 2 x - x^2)/(1 - 8 x + 10 x^2 + x^4), {x, 0, 30}], x] (* _Vincenzo Librandi_, Mar 15 2014 *)
%o A005389 (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
%o A005389 (SageMath)
%o A005389 def A005389_list(prec):
%o A005389 P.<x> = PowerSeriesRing(ZZ, prec)
%o A005389 return P( (1-2*x-x^2)/(1-8*x+10*x^2+x^4) ).list()
%o A005389 A005389_list(40) # _G. C. Greubel_, Nov 17 2022
%Y A005389 Bisection of A006864.
%K A005389 nonn,easy
%O A005389 1,2
%A A005389 _N. J. A. Sloane_, _Simon Plouffe_