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%I #27 Sep 08 2022 08:44:32
%S 6,327,11040,406731,14683587,532938234,19313709882,700238873703,
%T 25384576377228,920260033555854,33361543878231264,1209436952734137963,
%U 43844979765366718719,1589485790652572990865,57622670964678806649843,2088960040762846005003294
%N Number of 2-factors 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="/A003742/b003742.txt">Table of n, a(n) for n = 1..650</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_06">Index entries for linear recurrences with constant coefficients</a>, signature (26,396,-707,-6539,7239,-405).
%F a(1) = 6,
%F a(2) = 327,
%F a(3) = 11040,
%F a(4) = 406731,
%F a(5) = 14683587,
%F a(6) = 532938234, and
%F a(n) = 26a(n-1) + 396a(n-2) - 707a(n-3) - 6539a(n-4) + 7239a(n-5) - 405a(n-6).
%F G.f.: -3*x*(135*x^5 -2388*x^4 +1853*x^3 -54*x^2 -57*x -2)/(405*x^6 -7239*x^5 +6539*x^4 +707*x^3 -396*x^2 -26*x +1). - _Colin Barker_, Aug 30 2012
%t CoefficientList[Series[-3 (135 x^5 - 2388 x^4 + 1853 x^3 - 54 x^2 - 57 x - 2)/(405 x^6 - 7239 x^5 + 6539 x^4 + 707 x^3 - 396 x^2 - 26 x + 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 14 2013 *)
%t LinearRecurrence[{26,396,-707,-6539,7239,-405},{6,327,11040,406731,14683587,532938234},20] (* _Harvey P. Dale_, Dec 14 2015 *)
%o (Magma) I:=[6,327,11040,406731,14683587,532938234]; [n le 6 select I[n] else 26*Self(n-1)+396*Self(n-2)-707*Self(n-3)-6539*Self(n-4)+7239*Self(n-5)-405*Self(n-6): n in [1..20]]; // _Vincenzo Librandi_, Oct 14 2013
%o (PARI) Vec(-3*x*(135*x^5-2388*x^4+1853*x^3-54*x^2-57*x-2)/(405*x^6-7239*x^5+6539*x^4+707*x^3-396*x^2-26*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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