A290270 Number of minimal dominating sets in the n-wheel graph.

3, 4, 7, 6, 6, 15, 15, 22, 28, 45, 58, 79, 115, 159, 223, 307, 438, 609, 852, 1194, 1675, 2347, 3282, 4606, 6451, 9040, 12663, 17749, 24871, 34845, 48831, 68424, 95883, 134350, 188266, 263811, 369667, 518002, 725860, 1017129, 1425262, 1997179, 2798583

Offset: 3

Author

Eric W. Weisstein, Jul 25 2017

Keywords

• nonn

Comments

The n-wheel graph is well defined for n >= 4. If the sequence is extended to n=1 using A253413 then the initial terms are 1,2,3,4,... If the sequence is extended using the recurrence the initial terms are 7,1,3,4,... - Andrew Howroyd, Jul 27 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 3..1000
• Eric Weisstein's World of Mathematics, Minimal Dominating Set
• Eric Weisstein's World of Mathematics, Wheel Graph
• Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, -1, 1).

Crossrefs

Cf. A253413.

Programs

Magma

I:=[3,4,7,6,6,15,15,22,28]; [n le 9 select I[n] else Self(n-2)+Self(n-3)+Self(n-4)-Self(n-6)-1: n in [1..50]]; // Vincenzo Librandi, Aug 04 2017

Mathematica

Table[1 + RootSum[1 - #1^2 - #1^3 - #1^4 + #1^6 &, #^(n - 1) &], {n, 3, 20}] (* Eric W. Weisstein, Aug 04 2017 *)
LinearRecurrence[{1, 1, 0, 0, -1, -1, 1}, {3, 4, 7, 6, 6, 15, 15}, 20] (* Eric W. Weisstein, Aug 04 2017 *)
CoefficientList[Series[(3 + x - 5 x^3 - 7 x^4 + 6 x^5 + x^6)/((1 - x^2 - x^3 - x^4 + x^6) (1-x)), {x, 0,33}], x] (* Vincenzo Librandi, Aug 04 2017 *)

PARI

Vec(((7-6*x-5*x^2+2*x^5+x^6)) / ((1-x^2-x^3-x^4+x^6)*(1-x)) + O(x^40)) \\ Andrew Howroyd, Jul 27 2017

Formula

From Andrew Howroyd, Jul 27 2017: (Start)

a(n) = A253413(n-1) + 1 for n > 2.

a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-6) - 1 for n>8.

G.f.: x*(7 - 6*x - 5*x^2 + 2*x^5 + x^6) / ((1 - x^2 - x^3 -x^4 + x^6)*(1 - x)).

(End)

G.f.: x^3*(3+x-5*x^3-7*x^4+6*x^5+x^6)/((1-x^2-x^3-x^4+x^6)*(1-x)). - Vincenzo Librandi, Aug 04 2017

Extensions

a(3) and a(16)-a(45) from Andrew Howroyd, Jul 27 2017