A287594 Number of independent vertex sets in the n-helm graph.

3, 4, 12, 28, 72, 184, 480, 1264, 3360, 8992, 24192, 65344, 177024, 480640, 1307136, 3559168, 9699840, 26452480, 72173568, 196989952, 537802752, 1468536832, 4010582016, 10954043392, 29920862208, 81733033984, 223274237952, 609947435008, 1666309128192

Offset: 0

Author

Eric W. Weisstein, May 27 2017

Keywords

• nonn
• easy

Comments

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

Links

• Eric Weisstein's World of Mathematics, Helm Graph
• Eric Weisstein's World of Mathematics, Independent Vertex Set
• Index entries for linear recurrences with constant coefficients, signature (4, -2, -4).

Crossrefs

Cf. A080040.

Programs

Mathematica

Table[2^n + (1 - Sqrt[3])^n + (1 + Sqrt[3])^n, {n, 0, 20}] // Expand
Table[2^n + 2^(n/2) LucasL[n, Sqrt[2]], {n, 0, 20}] // Round
LinearRecurrence[{4, -2, -4}, {4, 12, 28}, {0, 20}]
CoefficientList[Series[(3 - 8 x + 2 x^2)/(1 - 4 x + 2 x^2 + 4 x^3), {x, 0, 20}], x]

Formula

a(n) = 2^n+A080040(n).

a(n) = 2^n+(1-sqrt(3))^n+(1+sqrt(3))^n.

a(n) = 4*a(n-1)-2*a(n-2)-4*a(n-3).

G.f.: (3-8*x+2*x^2)/((1-2*x)*(1-2*x-2*x^2)).