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A302488 Total domination number of the n X n grid graph.

%I A302488 #63 Aug 26 2025 04:51:15
%S A302488 0,1,2,3,6,9,12,15,20,25,30,35,42,49,56,63,72,81,90,99,110,121,132,
%T A302488 143,156,169,182,195,210,225,240,255,272,289,306,323,342,361,380,399,
%U A302488 420,441,462,483,506,529,552,575,600,625,650,675,702,729,756,783,812,841,870,899,930
%N A302488 Total domination number of the n X n grid graph.
%C A302488 Extended to a(0) and a(1) using the formula/recurrence. The total domination number of the 1 X 1 grid graph is undefined.
%H A302488 Andrew Howroyd, <a href="/A302488/b302488.txt">Table of n, a(n) for n = 0..1000</a>
%H A302488 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>.
%H A302488 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotalDominationNumber.html">Total Domination Number</a>.
%H A302488 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).
%F A302488 a(n) = ((-1)^n + 2*n*(n + 2) + 4*sin(n*Pi/2) - 1)/8.
%F A302488 a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6).
%F A302488 G.f.: x*(1 + 2*x^3 - x^4)/((1 - x)^3*(1 + x + x^2 + x^3)).
%F A302488 a(4*m + r) = (2*m + 1)*(2*m + r) for 0 <= r < 4. - _Charles Kusniec_, Aug 16 2025
%F A302488 From _Amiram Eldar_, Aug 26 2025: (Start)
%F A302488 Sum_{n>=1} 1/a(n) = Pi^2/8 + 3/2.
%F A302488 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/8 - 1/2. (End)
%t A302488 Table[(-1 + (-1)^n + 2 n (2 + n) + 4 Sin[n Pi/2])/8, {n, 0, 20}]
%t A302488 LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 1, 2, 3, 6, 9}, 20]
%t A302488 CoefficientList[Series[x (-1 - 2 x^3 + x^4)/((-1 + x)^3 (1 + x + x^2 + x^3)), {x, 0, 20}], x]
%o A302488 (PARI) for(n=0,30, print1(round(((-1)^n + 2*n*(n + 2) + 4*sin(n*Pi/2) - 1)/8), ", ")) \\ _G. C. Greubel_, Apr 09 2018
%o A302488 (PARI) a(n)=my(m=n\4); (2*m+1)*(2*m + n%4) \\ _Andrew Howroyd_, Aug 17 2025
%o A302488 (Magma) R:=RealField(); [Round(((-1)^n + 2*n*(n + 2) + 4*Sin(n*Pi(R)/2) - 1)/8): n in [0..30]]; // _G. C. Greubel_, Apr 09 2018
%Y A302488 Main diagonal of A300358.
%Y A302488 The four quadrasections are A002943, A016754, A002939(n+1), A000466(n+1).
%Y A302488 Bisections are A002378 and A085046.
%Y A302488 Cf. A303142.
%K A302488 nonn,easy,changed
%O A302488 0,3
%A A302488 _Eric W. Weisstein_, Apr 08 2018
%E A302488 a(0)=0 prepended and offset corrected by _Andrew Howroyd_, Aug 17 2025