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%I #41 Jan 01 2019 06:31:05
%S 5,1805,508805,140503005,38720000000,10668237057005,2939274449134805,
%T 809816405722655805,223117116976138566005,61472262298219520000000,
%U 16936571572967914651674005,4666290873812984282155907805,1285636259054921313298518442805
%N Number of spanning trees in C_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 P. Raff, <a href="/A003733/b003733.txt">Table of n, a(n) for n = 1..200</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 P. Raff, <a href="http://arxiv.org/abs/0809.2551">Spanning Trees in Grid Graphs</a>, arXiv:0809.2551 [math.CO], 2008. [Added by _Paul Raff_, Oct 30 2009]
%H P. Raff, <a href="http://www.math.rutgers.edu/~praff/span/5/12-13-24-35-45/index.xml">Analysis of the Number of Spanning Trees of C_5 x P_n.</a> Contains sequence, recurrence, generating function, and more. [Added by _Paul Raff_, Oct 30 2009] [broken link]
%H P. Raff, <a href="http://www.myraff.com/projects/spanning-trees-in-grid-graphs">Analysis of the Number of Spanning Trees of Grid Graphs</a>. [Added by _Paul Raff_, Oct 30 2009] [broken link]
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (319, -12441, 128319, -408001, 408801, -128319, 12441, -319, 1).
%F a(n) = 319*a(n-1) - 12441*a(n-2) + 128319*a(n-3) - 408001*a(n-4) + 408001*a(n-5) - 128319*a(n-6) + 12441*a(n-7) - 319*a(n-8) + a(n-9). [Modified by _Paul Raff_, Oct 30 2009]
%F G.f.: -5*x *(1+x) *(x^6+41*x^5-998*x^4+2722*x^3-998*x^2+41*x+1) / ( (x-1)*(x^4-279*x^3+961*x^2-279*x+1) *(x^4-39*x^3+281*x^2-39*x+1) ).
%F a(n) = 5 * (A143699(n))^2. - _R. K. Guy_, Mar 11 2010
%p a:= n-> (Matrix(1,9, (i,j)-> [0, 5, 1805, 508805, 140503005][1+abs(j-5)]). Matrix(9, (i,j)-> if (i=j-1) then 1 elif j=1 then -[408001, 128319, 12441, 319, 1][1/2+abs(i-9/2)] *(-1)^i else 0 fi)^n)[1, 5]: seq(a(n), n=1..20); # _Alois P. Heinz_, Mar 28 2009
%t a[n_] := (16/41)*Sinh[n*ArcCosh[(-9 - Sqrt[5])/4]]^2*Sinh[n*ArcCosh[(-9 + Sqrt[5])/4]]^2 // Round; Array[a, 20] (* _Jean-François Alcover_, Jan 31 2016, after _Peter Bala_ in A143699 *)
%Y Cf. A143699.
%K nonn
%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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