A382384 Number of minimum connected dominating sets in the n-Goldberg graph.

6, 96, 290, 744, 1974, 5376, 15642, 45480, 124014, 343008, 944658, 2596776, 7116390, 19409664, 52694730, 142812648, 385840030, 1039911520, 2796034626, 7501233256, 20084164374, 53677896192, 143214557050, 381504047912, 1014784646094, 2695617288672, 7151420301682

Offset: 3

Author

Eric W. Weisstein, Mar 23 2025

Keywords

• nonn
• easy

Comments

The connected domination number is given by 4*n - 1 = A004767(n - 1). - Andrew Howroyd, May 25 2025

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..500
• Eric Weisstein's World of Mathematics, Connected Dominating Set.
• Eric Weisstein's World of Mathematics, Goldberg Graph.
• Index entries for linear recurrences with constant coefficients, signature (4,2,-16,-7,20,24,-64,97,236,-246,-368,7,252,-772,-64,1920,0,-1024).

Crossrefs

Cf. A004767, A382431.

Programs

Mathematica

LinearRecurrence[{4, 2, -16, -7, 20, 24, -64, 97, 236, -246, -368, 7, 252, -772, -64, 1920, 0, -1024}, {6, 96, 290, 744, 1974, 5376, 15642, 45480, 124014, 343008, 944658, 2596776, 7116390, 19409664, 52694730, 142812648, 385840030, 1039911520}, 20] (* Eric W. Weisstein, Jun 04 2025 *)
CoefficientList[Series[2 (3 + 36 x - 53 x^2 - 256 x^3 - 2 x^4 + 592 x^5 + 1030 x^6 + 616 x^7 - 2817 x^8 - 2804 x^9 + 2591 x^10 + 2200 x^11 - 2592 x^12 - 2176 x^13 + 5168 x^14 + 3840 x^15 - 2304 x^16 - 2048 x^17)/((1 - x)^2 (1 + x)^2 (1 - x - 4 x^2)^2 (1 - x + 2 x^2)^2 (1 - x^2 - 4 x^3)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 04 2025 *)

Formula

G.f.: 2*x^3*(3 + 36*x - 53*x^2 - 256*x^3 - 2*x^4 + 592*x^5 + 1030*x^6 + 616*x^7 - 2817*x^8 - 2804*x^9 + 2591*x^10 + 2200*x^11 - 2592*x^12 - 2176*x^13 + 5168*x^14 + 3840*x^15 - 2304*x^16 - 2048*x^17)/((1 - x)*(1 + x)*(1 - x + 2*x^2)*(1 - x - 4*x^2)*(1 - x^2 - 4*x^3))^2. - Andrew Howroyd, May 25 2025

Extensions

a(7) onwards from Andrew Howroyd, May 24 2025