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A003731 Number of Hamiltonian cycles in C_5 X P_n.

%I A003731 #31 Feb 18 2025 17:38:36
%S A003731 1,5,30,160,850,4520,24040,127860,680040,3616880,19236840,102313600,
%T A003731 544168000,2894227280,15393318880,81871340160,435443220000,
%U A003731 2315960597120,12317733383040,65513444349760,348441653760640,1853231611930880,9856649945242240,52423856531251200
%N A003731 Number of Hamiltonian cycles in C_5 X P_n.
%D A003731 F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
%H A003731 Vincenzo Librandi, <a href="/A003731/b003731.txt">Table of n, a(n) for n = 1..1000</a>
%H A003731 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 A003731 F. Faase, <a href="http://www.iwriteiam.nl/counting.html">Counting Hamiltonian cycles in product graphs</a>
%H A003731 F. Faase, <a href="http://www.iwriteiam.nl/Cresults.html">Results from the counting program</a>
%H A003731 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4,2).
%F A003731 a(n) = 6*a(n-1) - 4*a(n-2) + 2*a(n-3) for n > 4.
%F A003731 G.f.: x*(1 - x + 4*x^2 - 2*x^3)/(1 - 6*x + 4*x^2 - 2*x^3). - _Colin Barker_, Sep 01 2012
%t A003731 CoefficientList[Series[(1 - x + 4 x^2 - 2 x^3)/(1 - 6 x + 4 x^2 - 2 x^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 14 2013 *)
%o A003731 (Magma) I:=[1,5,30,160]; [n le 4 select I[n] else 6*Self(n-1)-4*Self(n-2)+2*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Oct 14 2013
%o A003731 (PARI) a(n)=([0,1,0; 0,0,1; 2,-4,6]^(n-1)*[1;5;30])[1,1] \\ _Charles R Greathouse IV_, Jun 23 2020
%Y A003731 Column k=5 of A359855.
%K A003731 nonn,easy
%O A003731 1,2
%A A003731 _Frans J. Faase_
%E A003731 More terms from _Vincenzo Librandi_, Oct 14 2013