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Number of Hamiltonian paths in W_5 X P_n.
1

%I #41 Sep 08 2022 08:44:32

%S 24,1920,70184,2154592,58772296,2907860192,155789578368,8657124623872,

%T 486468905555968,27396295154262016,1543844945278500864,

%U 87013000712650358784,4904346018649432653824,276428194803652843536384,15580616335530275077357568,878187384790382283606458368,49498247371236620586614521856

%N Number of Hamiltonian paths in W_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="/A003738/b003738.txt">Table of n, a(n) for n = 1..580</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_04">Index entries for linear recurrences with constant coefficients</a>, signature (60,-128,-3328,-56832).

%F a(1) = 24, a(2) = 1920, a(3) = 70184, a(4) = 2154592, a(5) = 58772296, and a(n) = 60*a(n-1) - 128*a(n-2) - 3328*a(n-3) - 56832*a(n-4).

%F G.f.: 8*x*(3 +60*x -5243*x^2 -216352*x^3 -6720743*x^4)/(1 -60*x +128*x^2 +3328*x^3 +56832*x^4). - _Colin Barker_, Aug 31 2012

%p seq(coeff(series(8*x*(3 +60*x -5243*x^2 -216352*x^3 -6720743*x^4)/(1 -60*x +128*x^2 +3328*x^3 +56832*x^4), x, n+1), x, n), n = 1..40); # _G. C. Greubel_, Dec 25 2019

%t CoefficientList[Series[8*(3 +60*x -5243*x^2 -216352*x^3 -6720743*x^4)/(1 -60*x +128*x^2 +3328*x^3 +56832*x^4), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 14 2013 *)

%o (PARI) a(n) = if(n<1, 0, if(n<6, [24, 1920, 70184, 2154592, 58772296][n], 60*a(n-1) - 128*a(n-2) - 3328*a(n-3) - 56832*a(n-4) )); /* _Joerg Arndt_ and _Michael Somos_, Aug 30 2012 */

%o (Magma) I:=[24,1920,70184,2154592,58772296]; [n le 5 select I[n] else 60*Self(n-1)-128*Self(n-2)-3328*Self(n-3)-56832*Self(n-4): n in [1..20]]; // _Vincenzo Librandi_, Oct 14 2013

%o (Sage)

%o def A003738_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( 8*x*(3 +60*x -5243*x^2 -216352*x^3 -6720743*x^4)/(1 -60*x +128*x^2 +3328*x^3 +56832*x^4) ).list()

%o a=A003738_list(40); a[1:] # _G. C. Greubel_, Dec 25 2019

%o (GAP) a:=[1920,70184,2154592,58772296];; for n in [5..40] do a[n]:=60*a[n-1] -128*a[n-2]-3328*a[n-3]-56832*a[n-4]; od; Concatenation([24], a); # _G. C. Greubel_, Dec 25 2019

%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

%E Terms corrected by _Colin Barker_, Aug 30 2012