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

0,4

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, §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).

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

3rd column of A182797.

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

Sequence in context: A059491 A024279 A350873 * A232214 A012205 A156122

Adjacent sequences: A182785 A182786 A182787 * A182789 A182790 A182791

Keyword

nonn,easy

Author

Alois P. Heinz, Dec 02 2010

Status

approved