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%I #7 Dec 25 2019 15:31:00
%S 0,1,2,1,1,4,3,1,2,9,4,1,3,16,5,1,4,25,6,1,5,36,7,1,6,49,8,1,7,64,9,1,
%T 8,81,10,1,9,100,11,1,10,121,12,1,11,144,13,1,12,169,14,1,13,196,15,1,
%U 14,225,16,1,15,256,17,1,16,289,18,1,17,324,19,1,18,361,20,1
%N Number of minimum total dominating sets in the n-path graph.
%H Colin Barker, <a href="/A302654/b302654.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PathGraph.html">Path Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotalDominatingSet.html">Total Dominating Set</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) = ((-1)^n*(n - 2)^2 + (6 + n)^2 - 2*(n - 2)*(n + 6)*cos(n*Pi/2) - 48*sin(n*Pi/2))/6.
%F a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
%F G.f.: x^2*(1 + 2*x + x^2 + x^3 + x^4 - 3*x^5 - 2*x^6 - x^7 + x^9 + x^10) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3). - _Colin Barker_, Dec 25 2019
%t Table[Piecewise[{{1, Mod[n, 4] == 0}, {((n + 2)/4)^2, Mod[n, 4] == 2}, {(n - 1)/4, Mod[n, 4] == 1}, {(n + 5)/4, Mod[n, 4] == 3}}], {n, 20}]
%t Table[((-1)^n (n - 2)^2 + (6 + n)^2 - 2 (n - 2) (n + 6) Cos[n Pi/2] - 48 Sin[n Pi/2])/64, {n, 20}]
%t LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 1, 2, 1, 1, 4, 3, 1, 2, 9, 4, 1}, 20]
%o (PARI) concat(0, Vec(x^2*(1 + 2*x + x^2 + x^3 + x^4 - 3*x^5 - 2*x^6 - x^7 + x^9 + x^10) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3) + O(x^70))) \\ _Colin Barker_, Dec 25 2019
%K nonn,easy
%O 1,3
%A _Eric W. Weisstein_, Apr 11 2018
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