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

0,3

Comments

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

Links

Table of n, a(n) for n=0..59.

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).

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

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]

Prog

(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
(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

Crossrefs

Cf. A303210, A303213.

Sequence in context: A222264 A051783 A033083 * A328092 A242762 A005542

Adjacent sequences: A302403 A302404 A302405 * A302407 A302408 A302409

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 07 2018

Status

approved