A302404 Total domination number of the n-Moebius ladder.

0, 2, 2, 2, 3, 4, 4, 6, 6, 6, 7, 8, 8, 10, 10, 10, 11, 12, 12, 14, 14, 14, 15, 16, 16, 18, 18, 18, 19, 20, 20, 22, 22, 22, 23, 24, 24, 26, 26, 26, 27, 28, 28, 30, 30, 30, 31, 32, 32, 34, 34, 34, 35, 36, 36, 38, 38, 38, 39, 40, 40, 42, 42, 42, 43, 44, 44, 46, 46, 46, 47

Offset: 0

Author

Eric W. Weisstein, Apr 07 2018

Keywords

• nonn
• easy

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Moebius Ladder
• Eric Weisstein's World of Mathematics, Total Domination Number
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Crossrefs

Cf. A295420, A302405, A303046, A301337.

Programs

Magma

I:=[2,2,2,3,4,4,6]; [0] cat [n le 7 select I[n] else Self(n-1) + Self(n-6) - Self(n-7): n in [1..50]]; // G. C. Greubel, Apr 09 2018

Mathematica

Table[(3 - (-1)^n + 4 n + Cos[n Pi/3] - 3 Cos[2 n Pi/3] + Sqrt[3] Sin[n Pi/3] + Sin[2 n Pi/3]/Sqrt[3])/6, {n, 0, 20}]
LinearRecurrence[{1,0,0,0,0,1,-1}, {2,2,2,3,4,4,6}, {0, 50}]
CoefficientList[Series[x (2 + x^3 + x^4)/((-1 + x)^2 (1 + x + x^2 + x^3 + x^4 + x^5)), {x, 0, 20}], x]

PARI

x='x+O('x^50); concat(0, Vec(x*(2+x^3+x^4)/((1-x)^2*(1+x+x^2+x^3+x^4+x^5)))) \\ G. C. Greubel, Apr 09 2018

Formula

a(n) = (3 - (-1)^n + 4*n + cos(n*Pi/3) - 3*cos(2*n*Pi/3) + sqrt(3)*sin(n*Pi/3) + sin(2*n*Pi/3)/sqrt(3))/6.

a(n) = a(n-1) + a(n-6) - a(n-7).

G.f.: x*(2 + x^3 + x^4)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5)).

a(n) = a(n-6) + 4. - Andrew Howroyd, Apr 18 2018

a(n) = a(n-6*k) + 4*k. - Eric W. Weisstein, Apr 23 2018