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A304568 Number of minimum dominating sets in the n-antiprism graph.

%I A304568 #22 Feb 16 2025 08:33:54
%S A304568 2,4,15,12,5,40,14,140,45,5,154,24,546,98,5,384,34,1485,171,5,770,44,
%T A304568 3289,264,5,1352,54,6370,377,5,2170,64,11220,510,5,3264,74,18411,663,
%U A304568 5,4674,84,28595,836,5,6440,94,42504,1029,5,8602,104,60950,1242,5
%N A304568 Number of minimum dominating sets in the n-antiprism graph.
%C A304568 Sequence extrapolated to n=1 using formula. - _Andrew Howroyd_, May 20 2018
%H A304568 Andrew Howroyd, <a href="/A304568/b304568.txt">Table of n, a(n) for n = 1..1000</a>
%H A304568 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AntiprismGraph.html">Antiprism Graph</a>
%H A304568 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumDominatingSet.html">Minimum Dominating Set</a>
%H A304568 <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 A304568 From _Andrew Howroyd_, May 20 2018: (Start)
%F A304568 a(n) = 5*a(n-5) - 10*a(n-10) + 10*a(n-15) - 5*a(n-20) + a(n-25) for n > 25.
%F A304568 a(5*k) = 5, a(5*k+1) = 2*(5*k+1)*(k+1)*(2*k+3)/3, a(5*k+2) = 2*(5*k+2), a(5*k+3) = (5*k+3)*(2*k+5)*(2*k+4)*(2*k+3)/12, a(5*k+4)=(5*k+4)*(2*k+3). (End)
%F A304568 G.f.: x*(2 + 4*x + 15*x^2 + 12*x^3 + 5*x^4 + 30*x^5 - 6*x^6 + 65*x^7 - 15*x^8 - 20*x^9 - 26*x^10 - 6*x^11 - 4*x^12 - 7*x^13 + 30*x^14 - 6*x^15 + 14*x^16 + 5*x^17 + 11*x^18 - 20*x^19 - 6*x^21 - x^22 - x^23 + 5*x^24) / ((1 - x)^5*(1 + x + x^2 + x^3 + x^4)^5). - _Colin Barker_, May 22 2018
%t A304568 Table[Piecewise[{{5, Mod[n, 5] == 0}, {2 n (4 + n) (13 + 2 n)/75, Mod[n, 5] == 1}, {2 n, Mod[n, 5] == 2}, {n (7 + n) (9 + 2 n) (19 + 2 n)/750, Mod[n, 5] == 3}, {n (7 + 2 n)/5, Mod[n, 5] == 4}}], {n, 30}]
%t A304568 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}, {2, 4, 15, 12, 5, 40, 14, 140, 45, 5, 154, 24, 546, 98, 5, 384, 34, 1485, 171, 5, 770, 44, 3289, 264, 5}, 30]
%t A304568 CoefficientList[Series[(5 x^4)/(1 - x^5) + (2 x (2 + 3 x^5))/(-1 + x^5)^2 + (x^3 (-12 - 9 x^5 + x^10))/(-1 + x^5)^3 + (2 (1 + 16 x^5 + 3 x^10))/(-1 + x^5)^4 + (x^2 (-15 - 65 x^5 + 4 x^10 - 5 x^15 + x^20))/(-1 + x^5)^5, {x, 0, 30}], x]
%o A304568 (PARI) a(n)={[k->5, k->2*(5*k+1)*(k+1)*(2*k+3)/3, k->2*(5*k+2), k->(5*k+3)*(2*k+5)*(2*k+4)*(2*k+3)/12, k->(5*k+4)*(2*k+3)][n%5+1](n\5)} \\ _Andrew Howroyd_, May 20 2018
%o A304568 (PARI) Vec(x*(2 + 4*x + 15*x^2 + 12*x^3 + 5*x^4 + 30*x^5 - 6*x^6 + 65*x^7 - 15*x^8 - 20*x^9 - 26*x^10 - 6*x^11 - 4*x^12 - 7*x^13 + 30*x^14 - 6*x^15 + 14*x^16 + 5*x^17 + 11*x^18 - 20*x^19 - 6*x^21 - x^22 - x^23 + 5*x^24) / ((1 - x)^5*(1 + x + x^2 + x^3 + x^4)^5) + O(x^40)) \\ _Colin Barker_, May 22 2018
%Y A304568 Cf. A284699, A290377.
%K A304568 nonn,easy
%O A304568 1,1
%A A304568 _Eric W. Weisstein_, May 14 2018
%E A304568 a(1)-a(2) and terms a(21) and beyond from _Andrew Howroyd_, May 20 2018