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A182788
Number of n-colorings of the 3 X 3 X 3 triangular grid.
12
0, 0, 0, 6, 192, 1620, 7680, 26250, 72576, 172872, 368640, 721710, 1320000, 2283996, 3773952, 5997810, 9219840, 13770000, 20054016, 28564182, 39890880, 54734820, 73920000, 98407386, 129309312, 167904600, 215654400, 274218750, 345473856
OFFSET
0,4
COMMENTS
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.
REFERENCES
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.
FORMULA
a(n) = n*(n-1)*(n-2)^4.
G.f.: 6*x^3*(1 + 25*x + 67*x^2 + 27*x^3) / (1-x)^7.
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
MAPLE
a:= n-> n*(n-1)*(n-2)^4: seq(a(n), n=0..30);
MATHEMATICA
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 *)
PROG
(PARI) a(n)=n*(n-1)*(n-2)^4 \\ Charles R Greathouse IV, Jun 22 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Alois P. Heinz, Dec 02 2010
STATUS
approved