A182788 Number of n-colorings of the 3 X 3 X 3 triangular grid.

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

Author

Alois P. Heinz, Dec 02 2010

Keywords

• nonn
• easy

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, §1.10 Mathematics: Graph Theory 3, pp. 16-17.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Chromatic polynomial
• Wikipedia, Triangular grid graph
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

3rd column of A182797.

Cf. A178435, A182798, A182789, A182790, A182791, A182792, A182793, A182794, A182795, A182796.

Programs

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 *)

PARI

a(n)=n*(n-1)*(n-2)^4 \\ Charles R Greathouse IV, Jun 22 2016

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