A223363 6 X 6 X 6 triangular graph coloring a rectangular array: number of n X 1 0..20 arrays where 0..20 label nodes of the fully triangulated graph and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

21, 90, 420, 1992, 9552, 45984, 221760, 1070208, 5166336, 24943104, 120431616, 581486592, 2807648256, 13556490240, 65456455680, 316051587072, 1526031777792, 7368332673024, 35577456230400, 171783152467968, 829442428502016

Offset

1,1

Comments

Column 1 of A223370.

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Formula

Empirical: a(n) = 6*a(n-1) - 4*a(n-2) - 8*a(n-3).

Conjectures from Colin Barker, Aug 20 2018: (Start)

G.f.: 3*x*(7 - 12*x - 12*x^2) / ((1 - 2*x)*(1 - 4*x - 4*x^2)).

a(n) = (3/2)*(2^n-(2-2*sqrt(2))^n*(-1+sqrt(2)) + 2^(1/2+n)*(1+sqrt(2))^n + (2*(1+sqrt(2)))^n).

(End)

Example

Some solutions for n=3:
.20....2....4....4...18....4....6....7....6...17...14....4...11....8...15....5
.14....5....7....7...13....3....3....8....3...16...13....3...10....5...10....8
.20....2....8...11...14....1....4...12....6...11....9....6...16....2...11....5
Vertex neighbors:
0 -> 1 2
1 -> 0 2 3 4
2 -> 0 1 4 5
3 -> 1 4 6 7
4 -> 1 2 3 5 7 8
5 -> 2 4 8 9
6 -> 3 7 10 11
7 -> 3 4 6 8 11 12
8 -> 4 5 7 9 12 13
9 -> 5 8 13 14
10 -> 6 11 15 16
11 -> 6 7 10 12 16 17
12 -> 7 8 11 13 17 18
13 -> 8 9 12 14 18 19
14 -> 9 13 19 20
15 -> 10 16
16 -> 10 11 15 17
17 -> 11 12 16 18
18 -> 12 13 17 19
19 -> 13 14 18 20
20 -> 14 19

Crossrefs

Cf. A223370.

Sequence in context: A211464 A268257 A223370 * A284440 A020248 A225705

Adjacent sequences: A223360 A223361 A223362 * A223364 A223365 A223366

Keyword

nonn

Author

R. H. Hardin, Mar 19 2013

Status

approved