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A302405 Total domination number of the n-prism graph.

%I A302405 #16 Feb 16 2025 08:33:53
%S A302405 0,1,2,2,4,4,4,5,6,6,8,8,8,9,10,10,12,12,12,13,14,14,16,16,16,17,18,
%T A302405 18,20,20,20,21,22,22,24,24,24,25,26,26,28,28,28,29,30,30,32,32,32,33,
%U A302405 34,34,36,36,36,37,38,38,40,40,40,41,42,42,44,44,44,45,46,46,48
%N A302405 Total domination number of the n-prism graph.
%C A302405 Extended to a(0)-a(2) using the formula/recurrence.
%H A302405 G. C. Greubel, <a href="/A302405/b302405.txt">Table of n, a(n) for n = 0..10000</a>
%H A302405 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrismGraph.html">Prism Graph</a>
%H A302405 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotalDominationNumber.html">Total Domination Number</a>
%H A302405 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,1,-1).
%F A302405 a(n) = a(n-1) + a(n-6) - a(n-7).
%F A302405 G.f.: x*(1 + x + 2*x^3)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5)).
%F A302405 a(n) = a(n-6) + 4. - _Andrew Howroyd_, Apr 17 2018
%t A302405 Table[(3 + (-1)^n + 4 n - Cos[n Pi/3] - 3 Cos[2 n Pi/3] - Sqrt[3] Sin[n Pi/3] + Sin[2 n Pi/3]/Sqrt[3])/6, {n, 0, 50}]
%t A302405 LinearRecurrence[{1,0,0,0,0,1,-1}, {1,2,2,4,4,4,5,6}, {0, 50}]
%t A302405 CoefficientList[Series[x (1 + x + 2 x^3)/((-1 + x)^2 (1 + x + x^2 + x^3 + x^4 + x^5)), {x, 0, 50}], x]
%o A302405 (PARI) x='x+O('x^50); concat(0, Vec(x*(1+x+2*x^3)/((1-x)^2*(1+x+ x^2+x^3+ x^4+x^5)))) \\ _G. C. Greubel_, Apr 09 2018
%o A302405 (Magma) I:=[1,2,2,4,4,4,5,6]; [0] cat [n le 7 select I[n] else Self(n-1) + Self(n-6) - Self(n-7): n in [1..30]]; // _G. C. Greubel_, Apr 09 2018
%Y A302405 Cf. A296102, A303006, A303053.
%K A302405 nonn,easy
%O A302405 0,3
%A A302405 _Eric W. Weisstein_, Apr 07 2018