A364668 Domination and lower independence number of the n-Goldberg graph.

0, 3, 5, 7, 9, 11, 14, 16, 18, 20, 22, 25, 27, 29, 31, 33, 36, 38, 40, 42, 44, 47, 49, 51, 53, 55, 58, 60, 62, 64, 66, 69, 71, 73, 75, 77, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 102, 104, 106, 108, 110, 113, 115, 117, 119, 121, 124, 126, 128, 130, 132

Offset: 0

Author

Eric W. Weisstein, Aug 01 2023

Keywords

• nonn
• easy

Comments

The Goldberg graph is defined for n >= 3.

Extended to n = 0 through 2 using the formula/recurrence.

Disagrees with A195167(n) at n = 26, 31, 36, 41, ....

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Domination Number.
• Eric Weisstein's World of Mathematics, Goldberg Graph.
• Eric Weisstein's World of Mathematics, Lower Independence Number.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Crossrefs

Cf. A382431.

Programs

Mathematica

Table[(11 n - Cos[2 n Pi/5] - Cos[4 n Pi/5] + Sqrt[1 + 2/Sqrt[5]] Sin[2 n Pi/5] + Sqrt[1 - 2/Sqrt[5]] Sin[4 n Pi/5] + 2)/5, {n, 0, 20}]
LinearRecurrence[{1, 0, 0, 0, 1, -1}, {0, 3, 5, 7, 9, 11}, 20]
CoefficientList[Series[x (3 + 2 x + 2 x^2 + 2 x^3 + 2 x^4)/((-1 + x)^2 (1 + x + x^2 + x^3 + x^4)), {x, 0, 20}], x]

Formula

a(n) = a(n-1) + a(n-5) - a(n-6).

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

a(n) = floor((11*n + 4)/5). - Andrew Howroyd, May 25 2025

Extensions

Name extended by Eric W. Weisstein, Mar 10 2025