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A382431 Number of minimum dominating sets in the n-Goldberg graph.

%I A382431 #24 May 31 2025 09:34:41
%S A382431 63,12,5,1395,504,204,27,5,7370,1728,390,42,5,21052,3825,621,57,5,
%T A382431 46011,6930,897,72,5,86216,11178,1218,87,5,146041,16704,1584,102,5,
%U A382431 230265,23643,1995,117,5,344072,32130,2451,132,5,493051,42300,2952,147,5,683196,54288,3498,162,5
%N A382431 Number of minimum dominating sets in the n-Goldberg graph.
%C A382431 The domination number is given by A364668(n). - _Andrew Howroyd_, May 24 2025
%H A382431 Andrew Howroyd, <a href="/A382431/b382431.txt">Table of n, a(n) for n = 3..1000</a>
%H A382431 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbergGraph.html">Goldberg Graph</a>.
%H A382431 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumDominatingSet.html">Minimum Dominating Set</a>.
%H A382431 <a href="/index/Rec#order_25">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,5,0,0,0,0,-10,0,0,0,0,10,0,0,0,0,-5,0,0,0,0,1).
%F A382431 a(5*n) = 5.
%F A382431 a(5*n-1) = 3*(5*n-1); a(5*n-2) = 3*(5*n-2)*(3*n+11)/2; a(5*n-3) = 9*(5*n-3)*(n^2 + 11*n - 10)/2; a(5*n-4) = (5*n-4)*(27*n^3 + 594*n^2 + 9*n - 742)/8 for n > 2. - _Andrew Howroyd_, May 24 2025
%t A382431 Table[Piecewise[{{1395, n == 6}, {5, Mod[n, 5] == 0}, {3 n, Mod[n, 5] == 4}, {3 n (3 n + 61)/10, Mod[n, 5] == 3}, {9 n (n^2 + 61 n - 76)/50, Mod[n, 5] == 2}, {n (27 n^3 + 3294 n^2 + 25281 n - 42602)/1000, Mod[n, 5] == 1}}], {n, 3, 50}] (* _Eric W. Weisstein_, May 31 2025 *)
%t A382431 Join[{63, 12, 5, 1395}, LinearRecurrence[{0, 0, 0, 0, 5, 0, 0, 0, 0, -10, 0, 0, 0, 0, 10, 0, 0, 0, 0, -5, 0, 0, 0, 0, 1}, {504, 204, 27, 5, 7370, 1728, 390, 42, 5, 21052, 3825, 621, 57, 5, 46011, 6930, 897, 72, 5, 86216, 11178, 1218, 87, 5, 146041}, 50]] (* _Eric W. Weisstein_, May 31 2025 *)
%t A382431 CoefficientList[Series[-(1/((-1 + x)^5 (1 + x + x^2 + x^3 + x^4)^5)) (63 + 12 x + 5 x^2 + 1395 x^3 + 504 x^4 - 111 x^5 - 33 x^6 - 20 x^7 + 395 x^8 - 792 x^9 + 27 x^11 + 30 x^12 - 1848 x^13 + 225 x^14 + 81 x^15 - 3 x^16 - 20 x^17 + 501 x^18 + 45 x^19 - 33 x^20 - 3 x^21 + 5 x^22 - 44 x^23 + 18 x^24 + 6 x^28), {x, 0, 50}], x] (* _Eric W. Weisstein_, May 31 2025 *)
%o A382431 (PARI) a(n)=my(m=n\5+1,r=-n%5); if(r<=2, if(r==0, 5, 3*n*if(r==1, 1, (3*m+11)/2)), if(n==6, 1395, n*if(r==3, 9*(m^2 + 11*m - 10)/2, (27*m^3 + 594*m^2 + 9*m - 742)/8) )) \\ _Andrew Howroyd_, May 24 2025
%Y A382431 Cf. A364668 (domination number), A382384, A382657.
%K A382431 nonn,easy
%O A382431 3,1
%A A382431 _Eric W. Weisstein_, Mar 25 2025
%E A382431 a(11)-a(15) from _Eric W. Weisstein_, May 12 2025
%E A382431 a(16) onwards from _Andrew Howroyd_, May 24 2025