This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309380 #29 Sep 11 2019 19:42:27
%S A309380 180,240,1380,4200,15420,52080,177780,595320,1978860,6515520,21298980,
%T A309380 69168840,223369500,717772560,2296480980,7319252760,23247851340,
%U A309380 73615135200,232462779780,732245695080,2301319648380,7217727595440,22594530691380,70607719663800
%N A309380 Number of unordered pairs of 5-colorings of an n-wheel that differ in the coloring of exactly one vertex.
%C A309380 The n-wheel graph is defined for n >= 4. The value of a(3) was computed using the complete graph on 3 vertices.
%H A309380 Andrew Howroyd, <a href="/A309380/b309380.txt">Table of n, a(n) for n = 3..200</a>
%H A309380 Prateek Bhakta, Benjamin Brett Buckner, Lauren Farquhar, Vikram Kamat, Sara Krehbiel, Heather M. Russell, <a href="https://doi.org/10.1007/s00373-018-1985-6">Cut-Colorings in Coloring Graphs</a>, Graphs and Combinatorics, (2019) 35(1), 239-248.
%H A309380 Luis Cereceda, Janvan den Heuvel, Matthew Johnson, <a href="https://doi.org/10.1016/j.disc.2007.07.028">Connectedness of the graph of vertex-colourings</a>, Discrete Mathematics, (2008) 308(5-6), 913-919.
%H A309380 Aalok Sathe, <a href="https://github.com/aalok-sathe/coloring-graphs">Coloring Graphs Library</a>
%H A309380 Wikipedia, <a href="https://en.wikipedia.org/wiki/Wheel_graph">Wheel graph</a>
%H A309380 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6,-16,15,18).
%F A309380 From _Andrew Howroyd_, Sep 10 2019: (Start)
%F A309380 a(n) = 10*(2^(n-1) - 2*(-1)^n + (n-1)*(3^(n-2) - 3*(-1)^n)).
%F A309380 a(n) = 10*A092297(n-1) + 5*A326347(n-1).
%F A309380 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.
%F A309380 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.
%F A309380 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).
%F A309380 (End)
%o A309380 (PARI) a(n) = {10*(2^(n-1) - 2*(-1)^n + (n-1)*(3^(n-2) - 3*(-1)^n))} \\ _Andrew Howroyd_, Sep 10 2019
%o A309380 (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
%Y A309380 Cf. A092297, A106512, A309379 (similar sequence with 4 colors), A090860 (4-colorings), A309315 (5-colorings), A326347 (on n-cycle).
%K A309380 nonn
%O A309380 3,1
%A A309380 _Aalok Sathe_, Jul 26 2019
%E A309380 Terms a(12) and beyond from _Andrew Howroyd_, Sep 10 2019