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A302654 Number of minimum total dominating sets in the n-path graph.

%I A302654 #16 Jun 11 2025 00:59:06
%S A302654 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 A302654 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 A302654 14,225,16,1,15,256,17,1,16,289,18,1,17,324,19,1,18,361,20,1
%N A302654 Number of minimum total dominating sets in the n-path graph.
%H A302654 Colin Barker, <a href="/A302654/b302654.txt">Table of n, a(n) for n = 1..1000</a>
%H A302654 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumTotalDominatingSet.html">Minimum Total Dominating Set</a>.
%H A302654 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PathGraph.html">Path Graph</a>.
%H A302654 <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 A302654 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 A302654 a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
%F A302654 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
%F A302654 E.g.f.: ((12 + x^2)*cos(x) + (20 + 8*x + x^2)*cosh(x) + (5*x - 24)*sin(x) + (16 + 5*x)*sinh(x) - 32)/32. - _Stefano Spezia_, Jun 10 2025
%t A302654 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 A302654 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 A302654 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 A302654 (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
%Y A302654 Row 1 of A303293.
%Y A302654 Cf. A302653, A302655.
%K A302654 nonn,easy
%O A302654 1,3
%A A302654 _Eric W. Weisstein_, Apr 11 2018