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Total domination number of the n-ladder graph.
3

%I #12 May 02 2023 13:21:05

%S 0,2,2,2,4,4,4,6,6,6,8,8,8,10,10,10,12,12,12,14,14,14,16,16,16,18,18,

%T 18,20,20,20,22,22,22,24,24,24,26,26,26,28,28,28,30,30,30,32,32,32,34,

%U 34,34,36,36,36,38,38,38,40,40,40,42,42,42,44,44,44,46,46,46,48

%N Total domination number of the n-ladder graph.

%C Extended to a(0) using the formula/recurrence.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LadderGraph.html">Ladder 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_04">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 1, -1).

%F a(n) = 2*floor((n + 2)/3).

%F a(n) = 2/9*(3 + 3*n - 3*cos(2*n*Pi/3) + sqrt(3)*sin(2*n*Pi/3)).

%F a(n) = a(n-1) + a(n-3) - a(n-4).

%F G.f.: 2*x/((-1 + x)^2*(1 + x + x^2)).

%F a(n) = 2*A002264(n+2). - _R. J. Mathar_, May 02 2023

%t Table[2 Floor[(n + 2)/3], {n, 0, 20}]

%t 2 Floor[(Range[0, 20] + 2)/3]

%t Table[2/9 (3 + 3 n - 3 Cos[2 n Pi/3] + Sqrt[3] Sin[2 n Pi/3]), {n, 0, 20}]

%t LinearRecurrence[{1, 0, 1, -1}, {2, 2, 2, 4}, {0, 20}]

%t CoefficientList[Series[2 x/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x]

%K nonn,easy

%O 0,2

%A _Eric W. Weisstein_, Apr 07 2018