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

%I A364668 #13 May 25 2025 16:06:44
%S A364668 0,3,5,7,9,11,14,16,18,20,22,25,27,29,31,33,36,38,40,42,44,47,49,51,
%T A364668 53,55,58,60,62,64,66,69,71,73,75,77,80,82,84,86,88,91,93,95,97,99,
%U A364668 102,104,106,108,110,113,115,117,119,121,124,126,128,130,132
%N A364668 Domination and lower independence number of the n-Goldberg graph.
%C A364668 The Goldberg graph is defined for n >= 3.
%C A364668 Extended to n = 0 through 2 using the formula/recurrence.
%C A364668 Disagrees with A195167(n) at n = 26, 31, 36, 41, ....
%H A364668 Andrew Howroyd, <a href="/A364668/b364668.txt">Table of n, a(n) for n = 0..1000</a>
%H A364668 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominationNumber.html">Domination Number</a>.
%H A364668 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbergGraph.html">Goldberg Graph</a>.
%H A364668 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LowerIndependenceNumber.html">Lower Independence Number</a>.
%H A364668 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).
%F A364668 a(n) = a(n-1) + a(n-5) - a(n-6).
%F A364668 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)).
%F A364668 a(n) = floor((11*n + 4)/5). - _Andrew Howroyd_, May 25 2025
%t A364668 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}]
%t A364668 LinearRecurrence[{1, 0, 0, 0, 1, -1}, {0, 3, 5, 7, 9, 11}, 20]
%t A364668 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]
%Y A364668 Cf. A382431.
%K A364668 nonn,easy
%O A364668 0,2
%A A364668 _Eric W. Weisstein_, Aug 01 2023
%E A364668 Name extended by _Eric W. Weisstein_, Mar 10 2025