%I #11 May 13 2020 19:24:38

%S 16,2368,207496,15639936,1116199200,77643032832,5318859987584,

%T 360460090519552,24225364155392512,1617040095771160576,

%U 107318756823774554112,7087408485751290626048,466051677657117523779584,30530955397986792883159040,1993388935416599069605396480

%N Number of Eulerian cycles in the graph C_4 X C_n.

%C a(n) is divisible by 2^n.

%H Andrew Howroyd, <a href="/A298197/b298197.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerianCycle.html">Eulerian Cycle</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TorusGridGraph.html">Torus Grid Graph</a>

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (242, -24508, 1377576, -48319952, 1127281504, -18164303168, 206564578176, -1673816120832, 9654475382784, -39144748253184, 108479226544128, -194648732467200, 202715666841600, -92493840384000).

%F G.f.: 8*x*(2 - 188*x + 3321*x^2 + 177454*x^3 - 9041760*x^4 + 171251312*x^5 - 1590178736*x^6 + 5597941472*x^7 + 25225706112*x^8 - 343839085056*x^9 + 1466120669184*x^10 - 2913427243008*x^11 + 2262128394240*x^12)/((1 - 2*x)*(1 - 4*x)*(1 - 6*x)*(1 - 10*x)^2*(1 - 12*x)^2*(1 - 30*x)*(1 - 24*x + 64*x^2)*(1 - 66*x + 264*x^2)^2).

%F a(n) = 242*a(n-1) - 24508*a(n-2) + 1377576*a(n-3) - 48319952*a(n-4) + 1127281504*a(n-5) - 18164303168*a(n-6) + 206564578176*a(n-7) - 1673816120832*a(n-8) + 9654475382784*a(n-9) - 39144748253184*a(n-10) + 108479226544128*a(n-11) - 194648732467200*a(n-12) + 202715666841600*a(n-13) - 92493840384000*a(n-14). - _Eric W. Weisstein_, Jan 15 2018

%t Table[1/4 (-4 (12 - 4 Sqrt[5])^n - 4 (12 + 4 Sqrt[5])^n + 2^n (-3 + 3 2^(1 + n) + 3^n + 5^n - 15^n) + 2 (33 - 5 Sqrt[33])^n + 2 (33 + 5 Sqrt[33])^n) + 1/330 (-11 2^n (6 5^n + 5 6^n) + 50 (33 - 5 Sqrt[33])^n + 50 (33 + 5 Sqrt[33])^n) n, {n, 20}] // Expand (* _Eric W. Weisstein_, Jan 15 2018 *)

%t LinearRecurrence[{242, -24508, 1377576, -48319952, 1127281504, -18164303168, 206564578176, -1673816120832, 9654475382784, -39144748253184, 108479226544128, -194648732467200, 202715666841600, -92493840384000}, {16, 2368, 207496, 15639936, 1116199200, 77643032832, 5318859987584, 360460090519552, 24225364155392512, 1617040095771160576, 107318756823774554112, 7087408485751290626048, 466051677657117523779584, 30530955397986792883159040}, 20] (* _Eric W. Weisstein_, Jan 15 2018 *)

%t CoefficientList[Series[8 (2 - 188 x + 3321 x^2 + 177454 x^3 - 9041760 x^4 + 171251312 x^5 - 1590178736 x^6 + 5597941472 x^7 + 25225706112 x^8 - 343839085056 x^9 + 1466120669184 x^10 - 2913427243008 x^11 + 2262128394240 x^12)/((1 - 2 x) (1 - 4 x) (1 - 6 x) (1 - 10 x)^2 (1 - 12 x)^2 (1 - 30 x) (1 - 24 x + 64 x^2) (1 - 66 x + 264 x^2)^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Jan 15 2018 *)

%Y Row 4 of A298117.

%K nonn

%O 1,1

%A _Andrew Howroyd_, Jan 14 2018