A342285 Coordination sequence with respect to the central vertex of a dodecagon-based tiling of the plane by copies of a certain Goldberg quadrilateral tile.

1, 6, 6, 18, 30, 36, 42, 54, 60, 66, 72, 78, 84, 90, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348, 354, 360

Offset: 0

Author

N. J. A. Sloane, Mar 23 2021

Keywords

• nonn

Comments

There are many ways to tile the plane with the Goldberg tile; this is a particularly symmetric one.

In Cye Waldman's drawing, six copies of the gray sector are placed at the degree-4 vertices of the decagon, and 6 copies of a similar sector at the degree-6 vertices of the decagon.

References

• Goldberg, M. (1955). “Central Tessellations,” Scripta Mathematica, 21, pp. 253-260. See Fig.7b.
• Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987; Fig. 1.3.6(a), page 30.

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..2500
• Rémy Sigrist, Illustration of initial terms
• Rémy Sigrist, PARI program for A342285
• N. J. A. Sloane, Illustration of initial terms (For example, the a(3) = 18 vertices at 3 steps from the center are colored blue; the a(4) = 30 vertices at 4 steps from the center are green with a black circle.)
• Cye Waldman, The Goldberg Tile
• Cye Waldman, Central portion of tiling.
• Cye Waldman and others, Numerous postings to the Google Groups Tiling Mailing List about tilings with this Goldberg quadrilateral tile, March 2021.

Crossrefs

Cf. A138591, A250120.

Programs

PARI

See Links section.

Formula

Apparently, a(n) = 6*A138591(n-1) for n > 1. - Rémy Sigrist, Mar 30 2021

Extensions

More terms from Rémy Sigrist, Mar 29 2021