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A302946 Number of minimal (and minimum) total dominating sets in the 2n-crossed prism graph.

%I A302946 #31 Feb 16 2025 08:33:53
%S A302946 4,36,196,1156,6724,39204,228484,1331716,7761796,45239076,263672644,
%T A302946 1536796804,8957108164,52205852196,304278004996,1773462177796,
%U A302946 10336495061764,60245508192804,351136554095044,2046573816377476,11928306344169796,69523264248641316
%N A302946 Number of minimal (and minimum) total dominating sets in the 2n-crossed prism graph.
%C A302946 Extended to a(1) using the formula/recurrence.
%C A302946 Since minimal and minimum total dominating sets are equivalent, the crossed prism graphs could be said to be "well totally dominated".
%H A302946 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CrossedPrismGraph.html">Crossed Prism Graph</a>
%H A302946 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotalDominatingSet.html">Total Dominating Set</a>
%H A302946 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Well-CoveredGraph.html">Well-Covered Graph</a>
%H A302946 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,-1).
%F A302946 From _Andrew Howroyd_, Apr 16 2018: (Start)
%F A302946 a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3).
%F A302946 G.f.: 4*x*(1 + 4*x - x^2)/((1 + x)*(1 - 6*x + x^2)).
%F A302946 a(n) = 4*A090390(n) = 4*A001333(n)^2. (End)
%F A302946 a(n) = 2*(chebyshevT(n,3) + (-1)^n). - _Eric W. Weisstein_, Apr 17 2018
%F A302946 a(n) = 4*(-1)^n*chebyshevT(n,i)^2, where i is the imaginary unit. - _Eric W. Weisstein_, Apr 17 2018
%F A302946 E.g.f.: 2*(exp(-x) + exp(3*x)*cosh(2*sqrt(2)*x) - 2). - _Stefano Spezia_, Aug 03 2024
%t A302946 Table[2 (ChebyshevT[n, 3] + (-1)^n), {n, 20}]
%t A302946 Table[4 (-1)^n ChebyshevT[n, I]^2, {n, 20}]
%t A302946 LinearRecurrence[{5, 5, -1}, {4, 36, 196}, 20]
%t A302946 CoefficientList[Series[-4 (-1 - 4 x + x^2)/(1 - 5 x - 5 x^2 + x^3), {x, 0, 20}], x]
%o A302946 (PARI) Vec(4*(1 + 4*x - x^2)/((1 + x)*(1 - 6*x + x^2)) + O(x^30)) \\ _Andrew Howroyd_, Apr 16 2018
%o A302946 (PARI) a(n) = 2*(polchebyshev(n,1,3) + (-1)^n); \\ _Michel Marcus_, Apr 17 2018
%Y A302946 Cf. A001333, A002203 (sqrt), A090390 (quarter), A287062, A291772, A302941.
%K A302946 nonn,easy
%O A302946 1,1
%A A302946 _Eric W. Weisstein_, Apr 16 2018
%E A302946 a(1) and terms a(6) and beyond from _Andrew Howroyd_, Apr 16 2018