A309380 Number of unordered pairs of 5-colorings of an n-wheel that differ in the coloring of exactly one vertex.
180, 240, 1380, 4200, 15420, 52080, 177780, 595320, 1978860, 6515520, 21298980, 69168840, 223369500, 717772560, 2296480980, 7319252760, 23247851340, 73615135200, 232462779780, 732245695080, 2301319648380, 7217727595440, 22594530691380, 70607719663800
Offset: 3
Keywords
Links
- Andrew Howroyd, Table of n, a(n) for n = 3..200
- Prateek Bhakta, Benjamin Brett Buckner, Lauren Farquhar, Vikram Kamat, Sara Krehbiel, Heather M. Russell, Cut-Colorings in Coloring Graphs, Graphs and Combinatorics, (2019) 35(1), 239-248.
- Luis Cereceda, Janvan den Heuvel, Matthew Johnson, Connectedness of the graph of vertex-colourings, Discrete Mathematics, (2008) 308(5-6), 913-919.
- Aalok Sathe, Coloring Graphs Library
- Wikipedia, Wheel graph
- Index entries for linear recurrences with constant coefficients, signature (6,-6,-16,15,18).
Crossrefs
Programs
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PARI
a(n) = {10*(2^(n-1) - 2*(-1)^n + (n-1)*(3^(n-2) - 3*(-1)^n))} \\ Andrew Howroyd, Sep 10 2019 -
PARI
Vec(60*(3 - 14*x + 17*x^2 + 4*x^3 - 6*x^4)/((1 + x)^2*(1 - 2*x)*(1 - 3*x)^2) + O(x^30)) \\ Andrew Howroyd, Sep 10 2019
Formula
From Andrew Howroyd, Sep 10 2019: (Start)
a(n) = 10*(2^(n-1) - 2*(-1)^n + (n-1)*(3^(n-2) - 3*(-1)^n)).
a(n) = binomial(k, 2)*A106512(n-1, k-2) + k*(n-1)*(binomial(k-2, 2)*A106512(n-3, k-1) + binomial(k-3, 2)*A106512(n-2, k-1)) where k = 5.
a(n) = 6*a(n-1) - 6*a(n-2) - 16*a(n-3) + 15*a(n-4) + 18*a(n-5) for n > 7.
G.f.: 60*x^3*(3 - 14*x + 17*x^2 + 4*x^3 - 6*x^4)/((1 + x)^2*(1 - 2*x)*(1 - 3*x)^2).
(End)
Extensions
Terms a(12) and beyond from Andrew Howroyd, Sep 10 2019
Comments