A298022 Coordination sequence for Dual(3^3.4^2) tiling with respect to a trivalent node.

1, 3, 7, 12, 17, 23, 28, 33, 37, 42, 47, 51, 56, 61, 65, 70, 75, 79, 84, 89, 93, 98, 103, 107, 112, 117, 121, 126, 131, 135, 140, 145, 149, 154, 159, 163, 168, 173, 177, 182, 187, 191, 196, 201, 205, 210, 215, 219, 224, 229, 233, 238, 243, 247, 252, 257, 261

Offset

0,2

Comments

This tiling is also called the prismatic pentagonal tiling, or the cem-d net. It is one of the 11 Laves tilings.

References

B. Gruenbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987. See p. 96.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..1000

Chung, Ping Ngai, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, and Elena Wikner. Isoperimetric Pentagonal Tilings, Notices of the AMS 59, no. 5 (2012), pp. 632-640. See Fig. 1 (right).

Tom Karzes, Tiling Coordination Sequences

Frank Morgan, Optimal Pentagonal Tilings, Video, May 2021 [Mentions this tiling

Reticular Chemistry Structure Resource (RCSR), The cem-d tiling (or net)

Rémy Sigrist, PARI program for A298022

N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]

N. J. A. Sloane, Illustration of initial terms

Formula

Conjectures from Colin Barker, Jan 22 2018: (Start)

G.f.: (1 + 2*x + 4*x^2 + 4*x^3 + 3*x^4 + 2*x^5 - 2*x^8) / ((1 - x)^2*(1 + x + x^2)).

a(n) = a(n-1) + a(n-3) - a(n-4) for n>5.

(End)

Prog

(PARI) See Links section.

Crossrefs

See A298023 for partial sums, A298024 for a tetravalent point.

List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

Sequence in context: A350426 A332263 A310248 * A227133 A170883 A198463

Adjacent sequences: A298019 A298020 A298021 * A298023 A298024 A298025

Keyword

nonn

Author

N. J. A. Sloane, Jan 21 2018

Extensions

More terms from Rémy Sigrist, Jan 21 2018

Status

approved