A393349 Number of irredundant sets in the n-necklace graph.

6, 5, 39, 281, 1779, 11545, 75063, 487457, 3165795, 20561177, 133539239, 867300417, 5632879443, 36584012249, 237603155031, 1543167523681, 10022451115715, 65093079525529, 422761752924807, 2745721988266625, 17832713542963379, 115818598410317657, 752210128055638071

Offset

0,1

Comments

The sequence has been extended to a(0) using the recurrence. - Andrew Howroyd, Feb 12 2026

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

Eric Weisstein's World of Mathematics, Irredundant Set.

Eric Weisstein's World of Mathematics, Necklace Graph.

Index entries for linear recurrences with constant coefficients, signature (5,7,17,4,0,-8).

Formula

G.f.: (6 - 25*x - 28*x^2 - 51*x^3 - 8*x^4)/((1 + x + 2*x^2)*(1 - 6*x - 3*x^2 - 2*x^3 + 4*x^4)). - Andrew Howroyd, Feb 12 2026

Mathematica

Table[-(-1)^n 2^((n + 1)/2) Cos[(n - 1) ArcSec[2 Sqrt[2]]] - RootSum[4 - 2 # - 3 #^2 - 6 #^3 + #^4 &, -2 #^n - 3 #^(n + 1) - 6 #1^(n + 2) + #1^(n + 3) &]/4, {n, 20}] // FunctionExpand (* Eric W. Weisstein, Feb 14 2026 *)
LinearRecurrence[{5, 7, 17, 4, 0, -8}, {5, 39, 281, 1779, 11545, 75063}, {0, 20}] (* Eric W. Weisstein, Feb 14 2026 *)
CoefficientList[Series[(6 - 25 x - 28 x^2 - 51 x^3 - 8 x^4)/(1 - 5 x - 7 x^2 - 17 x^3 - 4 x^4 + 8 x^6), {x, 0, 20}], x] (* Eric W. Weisstein, Feb 14 2026 *)

Crossrefs

Cf. A392478.

Sequence in context: A288211 A038259 A358590 * A302750 A268000 A223529

Adjacent sequences: A393346 A393347 A393348 * A393350 A393351 A393352

Keyword

nonn,easy

Author

Eric W. Weisstein, Feb 12 2026

Extensions

a(0) prepended and a(9) onward from Andrew Howroyd, Feb 12 2026

Status

approved