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

0, 1, 2, 3, 6, 9, 12, 15, 20, 25, 30, 35, 42, 49, 56, 63, 72, 81, 90, 99, 110, 121, 132, 143, 156, 169, 182, 195, 210, 225, 240, 255, 272, 289, 306, 323, 342, 361, 380, 399, 420, 441, 462, 483, 506, 529, 552, 575, 600, 625, 650, 675, 702, 729, 756, 783, 812, 841, 870, 899, 930

Offset: 0

Author

Eric W. Weisstein, Apr 08 2018

Keywords

• nonn
• easy
• changed

Comments

Extended to a(0) and a(1) using the formula/recurrence. The total domination number of the 1 X 1 grid graph is undefined.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Grid Graph.
• Eric Weisstein's World of Mathematics, Total Domination Number.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Crossrefs

Main diagonal of A300358.

The four quadrasections are A002943, A016754, A002939(n+1), A000466(n+1).

Bisections are A002378 and A085046.

Cf. A303142.

Programs

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

Mathematica

Table[(-1 + (-1)^n + 2 n (2 + n) + 4 Sin[n Pi/2])/8, {n, 0, 20}]
LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 1, 2, 3, 6, 9}, 20]
CoefficientList[Series[x (-1 - 2 x^3 + x^4)/((-1 + x)^3 (1 + x + x^2 + x^3)), {x, 0, 20}], x]

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

PARI

a(n)=my(m=n\4); (2*m+1)*(2*m + n%4) \\ Andrew Howroyd, Aug 17 2025

Formula

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

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

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

a(4*m + r) = (2*m + 1)*(2*m + r) for 0 <= r < 4. - Charles Kusniec, Aug 16 2025

From Amiram Eldar, Aug 26 2025: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/8 + 3/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/8 - 1/2. (End)

Extensions

a(0)=0 prepended and offset corrected by Andrew Howroyd, Aug 17 2025