A302402 Total domination number of the n-ladder graph.

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, 18, 20, 20, 20, 22, 22, 22, 24, 24, 24, 26, 26, 26, 28, 28, 28, 30, 30, 30, 32, 32, 32, 34, 34, 34, 36, 36, 36, 38, 38, 38, 40, 40, 40, 42, 42, 42, 44, 44, 44, 46, 46, 46, 48

Offset: 0

Author

Eric W. Weisstein, Apr 07 2018

Keywords

• nonn
• easy

Comments

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

Links

• Eric Weisstein's World of Mathematics, Ladder Graph
• Eric Weisstein's World of Mathematics, Total Domination Number
• Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, -1).

Programs

Mathematica

Table[2 Floor[(n + 2)/3], {n, 0, 20}]
2 Floor[(Range[0, 20] + 2)/3]
Table[2/9 (3 + 3 n - 3 Cos[2 n Pi/3] + Sqrt[3] Sin[2 n Pi/3]), {n, 0, 20}]
LinearRecurrence[{1, 0, 1, -1}, {2, 2, 2, 4}, {0, 20}]
CoefficientList[Series[2 x/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x]

Formula

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

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

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

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

a(n) = 2*A002264(n+2). - R. J. Mathar, May 02 2023