A302255 Total domination number of the n-antiprism graph.

0, 1, 2, 2, 3, 4, 4, 4, 5, 6, 6, 7, 8, 8, 8, 9, 10, 10, 11, 12, 12, 12, 13, 14, 14, 15, 16, 16, 16, 17, 18, 18, 19, 20, 20, 20, 21, 22, 22, 23, 24, 24, 24, 25, 26, 26, 27, 28, 28, 28, 29, 30, 30, 31, 32, 32, 32, 33, 34, 34, 35, 36, 36, 36, 37, 38, 38, 39, 40, 40, 40, 41

Offset: 0

Author

Eric W. Weisstein, Apr 07 2018

Keywords

• nonn
• easy

Comments

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

Links

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

Crossrefs

Cf. A302652, A302760, A302763.

Programs

Magma

I:=[2,2,3,4,4,4,5,6]; [0,1] cat [n le 8 select I[n] else Self(n-1) + Self(n-7) - Self(n-8): n in [1..30]]; // G. C. Greubel, Apr 09 2018

Mathematica

Table[(4 + 4 n + E^(4 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 1] + E^(-4 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 2] + E^(-2 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 3] + E^(2 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 4] + E^(-6 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 5] + E^(6 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 6])/ 7, {n, 20}] // RootReduce
LinearRecurrence[{1,0,0,0,0,0,1,-1}, {1,2,2,3,4,4,4,5,6,6}, {0, 20}]
CoefficientList[Series[x (1 + x + x^3 + x^4)/((1 - x)^2 (1 + x + x^2 + x^3 + x^4 + x^5 + x^6)), {x, 0, 20}], x]

PARI

x='x+O('x^50); concat(0, Vec(x*(1+x+x^3+x^4)/((1-x)^2*(1+x+x^2+ x^3+x^4+x^5+x^6)))) \\ G. C. Greubel, Apr 09 2018

Formula

a(n) = a(n-1) + a(n-7) - a(n-8).

G.f.: x*(1 + x + x^3 + x^4)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).

a(n) = a(n-7) + 4. - Andrew Howroyd, Apr 18 2018

a(n) = a(n-7*k) + 4*k. - Eric W. Weisstein, Apr 19 2018