cp's OEIS Frontend

A347633 Number of minimum dominating sets in the path graph P_n.

%I A347633 #12 Feb 16 2025 08:34:02
%S A347633 1,2,1,4,3,1,8,4,1,13,5,1,19,6,1,26,7,1,34,8,1,43,9,1,53,10,1,64,11,1,
%T A347633 76,12,1,89,13,1,103,14,1,118,15,1,134,16,1,151,17,1,169,18,1,188,19,
%U A347633 1,208,20,1,229,21,1,251,22,1,274,23,1,298,24,1,323
%N A347633 Number of minimum dominating sets in the path graph P_n.
%H A347633 Andrew Howroyd, <a href="/A347633/b347633.txt">Table of n, a(n) for n = 1..1000</a>
%H A347633 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumDominatingSet.html">Minimum Dominating Set</a>
%H A347633 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PathGraph.html">Path Graph</a>
%H A347633 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,3,0,0,-3,0,0,1).
%F A347633 a(n) = 1               for n = 0 (mod 3)
%F A347633        (n^2+13*n+4)/18 for n = 1 (mod 3)
%F A347633        (n+4)/3         for n = 2 (mod 3).
%F A347633 a(n) = 3*a(n-3)-3*a(n-6)+a(n-9) for n > 9.
%F A347633 G.f.: -(x*(1+2*x+x^2+x^3-3*x^4-2*x^5-x^6+x^7+x^8))/((-1+x)^3*(1+x+x^2)^3).
%t A347633 Table[Piecewise[{{1, Mod[n, 3] == 0}, {(n^2 + 13 n + 4)/18, Mod[n, 3] == 1}, {(n + 4)/3, Mod[n, 3] == 2}}], {n, 20}]
%t A347633 LinearRecurrence[{0, 0, 3, 0, 0, -3, 0, 0, 1}, {1, 2, 1, 4, 3, 1, 8, 4, 1}, 20]
%t A347633 CoefficientList[Series[-(1 + 2 x + x^2 + x^3 - 3 x^4 - 2 x^5 - x^6 + x^7 + x^8)/((-1 + x)^3 (1 + x + x^2)^3), {x, 0, 20}], x]
%o A347633 (PARI) a(n)={if(n%3==0, 1, if(n%3==1, (n^2+13*n+4)/18,  (n+4)/3))} \\ _Andrew Howroyd_, Jan 18 2022
%Y A347633 Row 1 of A350815 and A350820.
%Y A347633 Cf. A347538, A347558, A350816.
%K A347633 nonn
%O A347633 1,2
%A A347633 _Eric W. Weisstein_, Sep 09 2021