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

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, 20, 21, 22, 24, 23, 24, 25, 27, 26, 27, 28, 30, 29, 30, 31, 33, 32, 33, 34, 36, 35, 36, 37, 39, 38, 39, 40, 42, 41, 42, 43

Offset: 3

Author

Aidan Johnson, Darren Narayan, and Andrew E. Vick, Jul 14 2022

Keywords

• nonn
• easy

Comments

P^3_n is the cube of path graph P_n.

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.

Links

• Georg Fischer, Table of n, a(n) for n = 3..1000
• Thomas Ansill, Bonnie Jacob, Jaime Penzellna, and Daniel Saavedra, Failed skew zero forcing on a graph, Linear Algebra and its Applications (2016) Vol. 509, 40-63.
• Aidan Johnson, Andrew Vick, Rigoberto Flórez, and Darren A. Narayan, Failed Skew Zero Forcing Numbers of Path Powers and Circulant Graphs, AppliedMath (2025) Vol. 5, No. 2, 32.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Programs

Mathematica

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 *)

Formula

a(n) = 3*floor((n-1)/4) + cos((n*Pi)/2) for n > 7.

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