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%I #25 Sep 08 2022 08:44:32
%S 6,204,4152,90012,1916640,41086080,878480688,18801691584,402251873664,
%T 8607198763968,184162678606848,3940493249544192,84313290894281472,
%U 1804026190433055744,38600161433802577920,825915469757893794816,17671849531462458580992
%N Number of Hamiltonian cycles in O_5 X P_n.
%D F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
%H Vincenzo Librandi, <a href="/A003743/b003743.txt">Table of n, a(n) for n = 1..750</a>
%H F. Faase, <a href="http://www.iwriteiam.nl/Cpaper.zip">On the number of specific spanning subgraphs of the graphs G X P_n</a>, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
%H F. Faase, <a href="http://www.iwriteiam.nl/counting.html">Counting Hamiltonian cycles in product graphs</a>
%H F. Faase, <a href="http://www.iwriteiam.nl/Cresults.html">Results from the counting program</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (16,136,-460,432,256).
%F a(1) = 6,
%F a(2) = 204,
%F a(3) = 4152,
%F a(4) = 90012,
%F a(5) = 1916640,
%F a(6) = 41086080, and
%F a(n) = 16a(n-1) + 136a(n-2) - 460a(n-3) + 432a(n-4) + 256a(n-5).
%F G.f.: 6*x*(256*x^5 -504*x^4 +234*x^3 -12*x^2 -18*x -1)/(256*x^5 +432*x^4 -460*x^3 +136*x^2 +16*x -1). - _Colin Barker_, Aug 30 2012
%t CoefficientList[Series[6 (256 x^5 - 504 x^4 + 234 x^3 - 12 x^2 - 18 x - 1)/(256 x^5 + 432 x^4 - 460 x^3 + 136 x^2 + 16 x - 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 14 2013 *)
%o (Magma) I:=[6,204,4152,90012,1916640,41086080]; [n le 6 select I[n] else 16*Self(n-1)+136*Self(n-2)-460*Self(n-3)+432*Self(n-4)+256*Self(n-5): n in [1..20]]; // _Vincenzo Librandi_, Oct 14 2013
%o (PARI) Vec(6*x*(256*x^5-504*x^4+234*x^3-12*x^2-18*x-1)/(256*x^5+432*x^4-460*x^3+136*x^2+16*x-1)+O(x^99)) \\ _Charles R Greathouse IV_, Jun 23 2020
%K nonn,easy
%O 1,1
%A _Frans J. Faase_
%E Added recurrence from Faase's web page. - _N. J. A. Sloane_, Feb 03 2009
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