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A355067 a(n) is the failed skew zero forcing number of P^3_n.

%I A355067 #62 Mar 28 2025 11:28:31
%S A355067 0,1,3,3,4,4,6,5,6,7,9,8,9,10,12,11,12,13,15,14,15,16,18,17,18,19,21,
%T A355067 20,21,22,24,23,24,25,27,26,27,28,30,29,30,31,33,32,33,34,36,35,36,37,
%U A355067 39,38,39,40,42,41,42,43
%N A355067 a(n) is the failed skew zero forcing number of P^3_n.
%C A355067 P^3_n is the cube of path graph P_n.
%C A355067 Given a graph G where each vertex is initially considered filled or unfilled, we apply the skew color change rule, which states that a vertex v becomes filled if and only if it is the unique empty neighbor of some other vertex in the graph. The failed skew zero forcing number of G, is the maximum cardinality of any subset S of vertices on which repeated application of the skew color change rule will not result in all vertices being filled.
%H A355067 Georg Fischer, <a href="/A355067/b355067.txt">Table of n, a(n) for n = 3..1000</a>
%H A355067 Thomas Ansill, Bonnie Jacob, Jaime Penzellna, and Daniel Saavedra, <a href="https://doi.org/10.1016/j.laa.2016.07.019">Failed skew zero forcing on a graph</a>, Linear Algebra and its Applications (2016) Vol. 509, 40-63.
%H A355067 Aidan Johnson, Andrew Vick, Rigoberto Flórez, and Darren A. Narayan, <a href="https://doi.org/10.3390/appliedmath5020032">Failed Skew Zero Forcing Numbers of Path Powers and Circulant Graphs</a>, AppliedMath (2025) Vol. 5, No. 2, 32.
%H A355067 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A355067 a(n) = 3*floor((n-1)/4) + cos((n*Pi)/2) for n > 7.
%F A355067 G.f.: x^3*(1 + 2*x + x^3 - x^4 - x^6 + x^8)/((1 - x)^2*(1 + x)*(1 + x^2)). - _Stefano Spezia_, Jul 15 2022
%t A355067 CoefficientList[Series[x^3*(1 + 2*x + x^3 - x^4 - x^6 + x^8)/((1 - x)^2*(1 + x)*(1 + x^2)), {x, 0, 59}], x][[3 ;; -1]] (* _Michael De Vlieger_, Mar 28 2025 *)
%K A355067 nonn,easy
%O A355067 3,3
%A A355067 _Aidan Johnson_, _Darren Narayan_, and _Andrew E. Vick_, Jul 14 2022