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A303046 Number of minimum total dominating sets in the n-Moebius ladder.

%I A303046 #24 Jun 10 2025 19:08:03
%S A303046 1,6,9,8,25,3,196,56,9,20,121,3,1521,154,9,32,289,3,5776,300,9,44,529,
%T A303046 3,15625,494,9,56,841,3,34596,736,9,68,1225,3,67081,1026,9,80,1681,3,
%U A303046 118336,1364,9,92,2209,3,194481,1750,9,104,2809,3,302500,2184,9
%N A303046 Number of minimum total dominating sets in the n-Moebius ladder.
%C A303046 Sequence extrapolated to n = 1 using recurrence. - _Andrew Howroyd_, Apr 18 2018
%H A303046 Andrew Howroyd, <a href="/A303046/b303046.txt">Table of n, a(n) for n = 1..200</a>
%H A303046 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumTotalDominatingSet.html">Minimum Total Dominating Set</a>.
%H A303046 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusLadder.html">Moebius Ladder</a>.
%H A303046 <a href="/index/Rec#order_30">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,5,0,0,0,0,0,-10,0,0,0,0,0,10,0,0,0,0,0,-5,0,0,0,0,0,1).
%F A303046 From _Andrew Howroyd_, Apr 18 2018: (Start)
%F A303046 a(n) = 5*a(n-6) - 10*a(n-12) + 10*a(n-18) - 5*a(n-24) + a(n-30) for n > 30.
%F A303046 a(6k) = 3, a(6k+1) = (6*k+1)^2*(k+1)^2, a(6k+2) = (6*k+2)*(4*k+3), a(6k+3) = 9, a(6k+4) = (6*k+4)*2, a(6k+5) = (6*k+5)^2. (End)
%F A303046 a(3k) = 6 - 3*(-1)^k. - _Eric W. Weisstein_, Apr 19 2018
%t A303046 Table[Piecewise[{{3, Mod[n, 6] == 0}, {(n (n + 5)/6)^2, Mod[n, 6] == 1}, {n (2 n + 5)/3, Mod[n, 6] == 2}, {9, Mod[n, 6] == 3}, {2 n, Mod[n, 6] == 4}, {n^2, Mod[n, 6] == 5}}], {n, 200}]
%t A303046 LinearRecurrence[{0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 1}, {1, 6, 9, 8, 25, 3, 196, 56, 9, 20, 121, 3, 1521, 154, 9, 32, 289, 3, 5776, 300, 9, 44, 529, 3, 15625, 494, 9, 56, 841, 3}, 200]
%t A303046 Rest @ CoefficientList[Series[3 x^6/(1 - x^6) - 9 x^3/(-1 + x^6) + 4 x^4 (2 + x^6)/(-1 + x^6)^2 - x^5 (25 + 46 x^6 + x^12)/(-1 + x^6)^3 - 2 x^2 (3 + 19 x^6 + 2 x^12)/(-1 + x^6)^3 - x (1 + 191 x^6 + 551 x^12 + 121 x^18)/(-1 + x^6)^5, {x, 0, 200}], x]
%o A303046 (PARI) a(n)=my(k=n\6,r=n%6);if(r<3, if(r==0, 3, if(r==1, n^2*(k+1)^2, n*(4*k+3))), if(r==3, 9, if(r==4, 2*n, n^2))) \\ _Andrew Howroyd_, Apr 18 2018
%Y A303046 Cf. A295420, A302404.
%K A303046 nonn,easy
%O A303046 1,2
%A A303046 _Eric W. Weisstein_, Apr 17 2018
%E A303046 a(1)-a(2) and terms a(14) and beyond from _Andrew Howroyd_, Apr 18 2018