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A282621 Number of Eulerian cycles in the graph C_3 X C_n.

%I A282621 #35 Feb 16 2025 08:33:41
%S A282621 8,320,8616,207496,4788808,108326760,2423906696,53891103656,
%T A282621 1193490502728,26367062410600,581618469479176,12817206071979816,
%U A282621 282280911579925448,6214413253138283240,136776355872474130056,3009909527048881143976,66229625352973066928968
%N A282621 Number of Eulerian cycles in the graph C_3 X C_n.
%H A282621 Andrew Howroyd, <a href="/A282621/b282621.txt">Table of n, a(n) for n = 1..200</a>
%H A282621 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerianCycle.html">Eulerian Cycle</a>
%H A282621 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TorusGridGraph.html">Torus Grid Graph</a>
%H A282621 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (47, -742, 4796, -13144, 12320).
%F A282621 a(n) = 47*a(n-1) - 742*a(n-2) + 4796*a(n-3) - 13144*a(n-4) + 12320*a(n-5) for n > 5.
%F A282621 G.f.: 8*x*(1 - 7*x - 61*x^2 + 202*x^3)/((1 - 2*x)*(1 - 4*x)*(1 - 5*x)*(1 - 14*x)*(1 - 22*x)).
%F A282621 a(n) = 22^n + (2^n + 3*2^(1 + 2*n) - 8*5^n - 2^(1 + n)*7^n)/3. - _Eric W. Weisstein_, Jan 15 2018
%t A282621 Table[22^n + 1/3 (2^n + 3 2^(1 + 2 n) - 8 5^n - 2^(1 + n) 7^n), {n, 20}] (* _Eric W. Weisstein_, Jan 15 2018 *)
%t A282621 LinearRecurrence[{47, -742, 4796, -13144, 12320}, {8, 320, 8616, 207496, 4788808}, 20] (* _Eric W. Weisstein_, Jan 15 2018 *)
%t A282621 CoefficientList[Series[8 (1 - 7 x - 61 x^2 + 202 x^3)/((1 - 2 x) (1 - 4 x) (1 - 5 x) (1 - 14 x) (1 - 22 x)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Jan 15 2018 *)
%o A282621 (PARI) Vec(8*(1 - 7*x - 61*x^2 + 202*x^3)/((1 - 2*x)*(1 - 4*x)*(1 - 5*x)*(1 - 14*x)*(1 - 22*x)) + O(x^20))
%Y A282621 Row 3 of A298117.
%K A282621 nonn
%O A282621 1,1
%A A282621 _Andrew Howroyd_, Jan 14 2018