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A382657 Number of minimum total dominating sets in the n-Goldberg graph.

%I A382657 #20 May 28 2025 00:53:35
%S A382657 16,277,10,2386,28,33301,360,10,4334,60,67288,728,10,9856,102,150750,
%T A382657 1292,10,19222,154,299368,2124,10,34112,216,549276,3306,10,56730,288,
%U A382657 951456,4930,10,89916,370,1576202,7098,10,137268,462,2518596,9922,10,203274,564,3905148,13524,10
%N A382657 Number of minimum total dominating sets in the n-Goldberg graph.
%C A382657 For n > 3, the total domination number is given by 2*floor((7*n+4)/5) = 2*A047332(n+1). For n = 3, the total domination number is 9. - _Andrew Howroyd_, May 24 2025
%H A382657 Andrew Howroyd, <a href="/A382657/b382657.txt">Table of n, a(n) for n = 3..1000</a>
%H A382657 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbergGraph.html">Goldberg Graph</a>.
%H A382657 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumTotalDominatingSet.html">Minimum Total Dominating Set</a>.
%H A382657 <a href="/index/Rec#order_45">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,9, 0,0,0,0,-36, 0,0,0,0,84, 0,0,0,0,-126, 0,0,0,0,126, 0,0,0,0,-84, 0,0,0,0,36, 0,0,0,0,-9, 0,0,0,0,1).
%F A382657 a(5*n) = 10.
%F A382657 From _Andrew Howroyd_, May 24 2025: (Start)
%F A382657 a(5*n-1) = (5*n-1)*(n + 13)*(n^2 + 2*n + 24)/12 for n >= 2;
%F A382657 a(5*n-2) = (5*n-2)*(n^7 + 70*n^6 + 1750*n^5 + 39340*n^4 + 525889*n^3 + 2944270*n^2 + 15922920*n + 12216960)/20160 for n >= 3;
%F A382657 a(5*n-3) = (5*n-3)*(n + 2) for n >= 2;
%F A382657 a(5*n-4) = (5*n-4)*(n^5 + 35*n^4 + 365*n^3 + 5485*n^2 + 21114*n + 16200)/360 for n >= 3. (End)
%o A382657 (PARI) a(n) = { my(m=n\5+1,r=-n%5); if(n<=8, [16, 277, 10, 2386, 28, 33301][n-2], if(r<=2, if(r==0, 10, n*if(r==1, (m + 13)*(m^2 + 2*m + 24)/12, (m^7 + 70*m^6 + 1750*m^5 + 39340*m^4 + 525889*m^3 + 2944270*m^2 + 15922920*m + 12216960)/20160)), n*if(r==3, (m + 2), (m^5 + 35*m^4 + 365*m^3 + 5485*m^2 + 21114*m + 16200)/360) )) } \\ _Andrew Howroyd_, May 24 2025
%Y A382657 Cf. A382431.
%K A382657 nonn,easy
%O A382657 3,1
%A A382657 _Eric W. Weisstein_, Apr 02 2025
%E A382657 a(8)-a(12) from _Eric W. Weisstein_, May 11 2025
%E A382657 a(13) onwards from _Andrew Howroyd_, May 24 2025