A265045 Coordination sequence for a 6.6.6 point in the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6} of the plane by squares and dominoes (hexagons).

1, 3, 7, 11, 14, 18, 23, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232

Offset: 0

Author

N. J. A. Sloane and Susanna Cuyler, Dec 27 2015

Keywords

• nonn
• easy

Comments

This tiling is 3-transitive but not 3-uniform since the polygons are not regular. It is a common floor-tiling.

The coordination sequences with respect to the points of types 4.6.6 (labeled "C" in the illustration), 6.6.6 ("B"), 6.6.6.6 ("A") are A265046, A265045, and A008574, respectively. The present sequence is for a "B" point.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, A portion of the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6}
• N. J. A. Sloane, A portion of the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6} showing the three types of point
• N. J. A. Sloane, Hand-drawn illustration showing a(0) to a(10)
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A008574, A265046.

Programs

Mathematica

LinearRecurrence[{2,-1},{1,3,7,11,14,18,23,28,32},60] (* Harvey P. Dale, Sep 23 2017 *)

PARI

Vec((1+x)*(1+2*x^2-2*x^3+x^4+x^6-x^7)/(1-x)^2 + O(x^100)) \\ Colin Barker, Jan 01 2016

Formula

For n >= 7 all three sequences equal 4n. (For n >= 7 the n-th shell contains n-1 points in the interior of each quadrant plus 4 points on the axes.)

From Colin Barker, Jan 01 2016: (Start)

a(n) = 2*a(n-1)-a(n-2) for n>8.

a(n) = 4*n for n>6.

G.f.: (1+x)*(1+2*x^2-2*x^3+x^4+x^6-x^7) / (1-x)^2.

(End)