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A326347 Number of unordered pairs of 4-colorings of an n-cycle that differ in the coloring of exactly one vertex.

%I A326347 #13 Feb 16 2025 08:33:58
%S A326347 36,240,780,2952,10164,35040,118044,393720,1299012,4251600,13817388,
%T A326347 44641128,143488980,459165120,1463588412,4649045976,14721978468,
%U A326347 46490458800,146444944716,460255541064,1443528741876,4518872583840,14121476823900,44059007691192
%N A326347 Number of unordered pairs of 4-colorings of an n-cycle that differ in the coloring of exactly one vertex.
%H A326347 Andrew Howroyd, <a href="/A326347/b326347.txt">Table of n, a(n) for n = 3..200</a>
%H A326347 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>
%H A326347 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,2,-12,-9).
%F A326347 a(n) = n*(3*A218034(n-2) + A218034(n-1)).
%F A326347 a(n) = 6*n*(3^(n-2) + (-1)^n).
%F A326347 a(n) = 12*n*A046717(n-2).
%F A326347 a(n) = 4*a(n-1) + 2*a(n-2) - 12*a(n-3) - 9*a(n-4) for n > 6.
%F A326347 G.f.: 12*x^3*(3 + 8*x - 21*x^2 - 18*x^3)/((1 + x)^2*(1 - 3*x)^2).
%o A326347 (PARI) a(n) = 6*n*(3^(n-2) + (-1)^n);
%Y A326347 Cf. A046717, A218034, A309380.
%K A326347 nonn,easy
%O A326347 3,1
%A A326347 _Andrew Howroyd_, Sep 11 2019