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

0, 1, 2, 3, 4, 8, 10, 14, 16, 23, 26, 33, 36, 46, 50, 60, 64, 77, 82, 95, 100, 116, 122, 138, 144, 163, 170, 189, 196, 218, 226, 248, 256, 281, 290, 315, 324, 352, 362, 390, 400, 431, 442, 473, 484, 518, 530, 564, 576, 613, 626, 663, 676, 716, 730, 770, 784, 827, 842, 885

Offset: 0

Author

Eric W. Weisstein, Apr 07 2018

Keywords

• nonn
• easy

Comments

Extended to a(0)-a(2) using the formula/recurrence.

Links

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

Crossrefs

Cf. A303210, A303213.

Programs

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

Mathematica

Table[(3-(-1)^n*(n-1)+n+2*n^2-4*Cos[n*Pi/2]+2*Sin[n*Pi/2])/8, {n, 0, 20}]
LinearRecurrence[{1, 1, -1, 1, -1, -1, 1}, {1, 2, 3, 4, 8, 10, 14}, {0, 20}]
CoefficientList[Series[-x (1 + x + 2 x^4)/((-1 + x)^3 (1 + x)^2 (1 + x^2)), {x, 0, 20}], x]

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

Formula

a(n) = (3 -(-1)^n*(n - 1) + n + 2*n^2 - 4*cos(n*Pi/2) + 2*sin(n*Pi/2))/8.

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7).

G.f.: -x*(1 + x + 2*x^4)/((-1 + x)^3*(1 + x)^2*(1 + x^2)).

a(n) ~ n^2/4. - Andrew Howroyd, Apr 21 2018