A355399 a(n) is the failed skew zero forcing number of C^2_n.

0, 1, 2, 4, 3, 4, 6, 5, 6, 8, 6, 8, 10, 8, 10, 12, 10, 12, 14, 12, 14, 16, 14, 16, 18, 16, 18, 20, 18, 20, 22, 20, 22, 24, 22, 24, 26, 24, 26, 28, 26, 28, 30, 28, 30, 32, 30, 32, 34, 32, 34, 36, 34, 36, 38, 36, 38, 40, 38, 40, 42, 40, 42, 44, 42, 44, 46, 44, 46

Offset: 3

Author

Darren Narayan, Andrew E. Vick, and Aidan Johnson, Jun 30 2022

Keywords

• nonn
• easy

Comments

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 color change rule will not result in all vertices being filled. Note that C^2_n = Ci_n(1,2) is the square of C_n.

Links

• Thomas Ansill, Bonnie Jacob, Jaime Penzellna, and Daniel Saavedra, Failed skew zero forcing on a graph, Linear Algebra and its Applications, vol. 509 (2016), 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 (1,0,1,-1).

Crossrefs

Cf. A008611, A343648.

Formula

a(n) = 2*floor(n/3) + 2*(ceiling(n/(3*floor(n/3) + 1)) - floor(n/(3*floor(n/3) +1 )) - 1) for n >= 11.

a(n) = 2*A008611(n-3) for n >= 11.

Extensions

More terms from Stefano Spezia, Jun 30 2022