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A302406 Total domination number of the n X n torus grid graph.

%I A302406 #19 Feb 16 2025 08:33:53
%S A302406 0,1,2,3,4,8,10,14,16,23,26,33,36,46,50,60,64,77,82,95,100,116,122,
%T A302406 138,144,163,170,189,196,218,226,248,256,281,290,315,324,352,362,390,
%U A302406 400,431,442,473,484,518,530,564,576,613,626,663,676,716,730,770,784,827,842,885
%N A302406 Total domination number of the n X n torus grid graph.
%C A302406 Extended to a(0)-a(2) using the formula/recurrence.
%H A302406 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TorusGridGraph.html">Torus Grid Graph</a>
%H A302406 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotalDominationNumber.html">Total Domination Number</a>
%H A302406 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1,-1,1).
%F A302406 a(n) = (3 -(-1)^n*(n - 1) + n + 2*n^2 - 4*cos(n*Pi/2) + 2*sin(n*Pi/2))/8.
%F A302406 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7).
%F A302406 G.f.: -x*(1 + x + 2*x^4)/((-1 + x)^3*(1 + x)^2*(1 + x^2)).
%F A302406 a(n) ~ n^2/4. - _Andrew Howroyd_, Apr 21 2018
%t A302406 Table[(3-(-1)^n*(n-1)+n+2*n^2-4*Cos[n*Pi/2]+2*Sin[n*Pi/2])/8, {n, 0, 20}]
%t A302406 LinearRecurrence[{1, 1, -1, 1, -1, -1, 1}, {1, 2, 3, 4, 8, 10, 14}, {0, 20}]
%t A302406 CoefficientList[Series[-x (1 + x + 2 x^4)/((-1 + x)^3 (1 + x)^2 (1 + x^2)), {x, 0, 20}], x]
%o A302406 (PARI) for(n=0,30, print1(round((3-(-1)^n*(n-1) +n +2*n^2 -4*cos(n*Pi/2) + 2*sin(n*Pi/2))/8), ", ")) \\ _G. C. Greubel_, Apr 09 2018
%o A302406 (Magma) R:=RealField(); [Round((3 -(-1)^n*(n-1) +n +2*n^2 - 4*Cos(n*Pi(R)/2) + 2*Sin(n*Pi(R)/2))/8): n in [0..20]]; // _G. C. Greubel_, Apr 09 2018
%Y A302406 Cf. A303210, A303213.
%K A302406 nonn,easy
%O A302406 0,3
%A A302406 _Eric W. Weisstein_, Apr 07 2018