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

0,2

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).

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)

Mathematica

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

Prog

(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

Crossrefs

Cf. A008574, A265046.

Sequence in context: A246170 A190694 A310206 * A310207 A189385 A310208

Adjacent sequences: A265042 A265043 A265044 * A265046 A265047 A265048

Keyword

nonn,easy

Author

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

Status

approved