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Number of minimum dominating sets in the n-Moebius ladder.
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%I #16 Feb 16 2025 08:34:02

%S 9,24,10,4,14,80,18,4,22,168,26,4,30,288,34,4,38,440,42,4,46,624,50,4,

%T 54,840,58,4,62,1088,66,4,70,1368,74,4,78,1680,82,4,86,2024,90,4,94,

%U 2400,98,4,102,2808,106,4,110,3248,114,4,118,3720,122,4,126,4224

%N Number of minimum dominating sets in the n-Moebius ladder.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumDominatingSet.html">Minimum Dominating Set</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusLadder.html">Moebius Ladder</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

%F a(n) = n*(n+2) for n == 0 (mod 4).

%F a(n) = 2*n for n == 1 (mod 2) and n > 3.

%F a(n) = 4 for n == 2 (mod 4).

%F a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12) for n > 3.

%F G.f.: x^3*(-9 - 24*x - 10*x^2 - 4*x^3 + 13*x^4 - 8*x^5 + 12*x^6 + 8*x^7 - 7*x^8 - 2*x^10 - 4*x^11 + 3*x^12)/((-1 + x)^3*(1 + x)^3*(1 + x^2)^3).

%t Table[Piecewise[{{9, n == 3}, {n (n + 2), Mod[n, 4] == 0}, {2 n, Mod[n, 2] == 1}, {4, Mod[n, 4] == 2}}, 0], {n, 3, 20}]

%t Join[{9}, LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {24, 10, 4, 14, 80, 18, 4, 22, 168, 26, 4, 30}, 20]]

%t CoefficientList[Series[(-9 - 24 x - 10 x^2 - 4 x^3 + 13 x^4 - 8 x^5 + 12 x^6 + 8 x^7 - 7 x^8 - 2 x^10 - 4 x^11 + 3 x^12)/((-1 + x)^3 (1 + x)^3 (1 + x^2)^3), {x, 0, 20}], x]

%Y Cf. A284663, A290337.

%K nonn,easy,changed

%O 3,1

%A _Eric W. Weisstein_, Sep 06 2021