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%I #14 Sep 08 2022 08:46:21
%S 1,2,3,6,9,12,15,20,25,30,35,42,49,56,63,72,81,90,99,110,121,132,143,
%T 156,169,182,195,210,225,240,255,272,289,306,323,342,361,380,399,420,
%U 441,462,483,506,529,552,575,600,625,650,675,702,729,756,783,812,841,870,899,930
%N Total domination number of the n X n grid graph.
%C Extended to a(1) using the formula/recurrence.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotalDominationNumber.html">Total Domination Number</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).
%F a(n) = ((-1)^n + 2*n*(n + 2) + 4*sin(n*Pi/2) - 1)/8.
%F a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6).
%F G.f.: (1 + 2*x^3 - x^4)/((1 - x)^3*(1 + x + x^2 + x^3)).
%t Table[(-1 + (-1)^n + 2 n (2 + n) + 4 Sin[n Pi/2])/8, {n, 20}]
%t LinearRecurrence[{2, -1, 0, 1, -2, 1}, {1, 2, 3, 6, 9, 12}, 20]
%t CoefficientList[Series[(-1 - 2 x^3 + x^4)/((-1 + x)^3 (1 + x + x^2 + x^3)), {x, 0, 20}], x]
%o (PARI) for(n=1,30, print1(round(((-1)^n + 2*n*(n + 2) + 4*sin(n*Pi/2) - 1)/8), ", ")) \\ _G. C. Greubel_, Apr 09 2018
%o (Magma) R:=RealField(); [Round(((-1)^n + 2*n*(n + 2) + 4*Sin(n*Pi(R)/2) - 1)/8): n in [1..30]]; // _G. C. Greubel_, Apr 09 2018
%Y Main diagonal of A300358.
%Y Cf. A303142.
%K nonn
%O 2,2
%A _Eric W. Weisstein_, Apr 08 2018