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%I #37 Aug 01 2023 14:33:55
%S 0,0,0,6,192,1620,7680,26250,72576,172872,368640,721710,1320000,
%T 2283996,3773952,5997810,9219840,13770000,20054016,28564182,39890880,
%U 54734820,73920000,98407386,129309312,167904600,215654400,274218750,345473856
%N Number of n-colorings of the 3 X 3 X 3 triangular grid.
%C The 3 X 3 X 3 triangular grid has 3 rows with k vertices in row k. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The graph has 6 vertices and 9 edges altogether.
%D Burkard Polster and Marty Ross, Math Goes to the Movies, The Johns Hopkins University Press, Baltimore, 2013, ยง1.10 Mathematics: Graph Theory 3, pp. 16-17.
%H Alois P. Heinz, <a href="/A182788/b182788.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic polynomial</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = n*(n-1)*(n-2)^4.
%F G.f.: 6*x^3*(1 + 25*x + 67*x^2 + 27*x^3) / (1-x)^7.
%F a(0)=0, a(1)=0, a(2)=0, a(3)=6, a(4)=192, a(5)=1620, a(6)=7680, a(n) = 7*a(n-1) -21*a(n-2) +35*a(n-3) -35*a(n-4) +21*a(n-5) -7*a(n-6) +a(n-7). - _Harvey P. Dale_, Dec 10 2011
%p a:= n-> n*(n-1)*(n-2)^4: seq(a(n), n=0..30);
%t Table[n(n-1)(n-2)^4,{n,0,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,0,0,6,192,1620,7680},30] (* _Harvey P. Dale_, Dec 10 2011 *)
%o (PARI) a(n)=n*(n-1)*(n-2)^4 \\ _Charles R Greathouse IV_, Jun 22 2016
%Y 3rd column of A182797.
%Y Cf. A178435, A182798, A182789, A182790, A182791, A182792, A182793, A182794, A182795, A182796.
%K nonn,easy
%O 0,4
%A _Alois P. Heinz_, Dec 02 2010
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