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
Keywords
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
Crossrefs
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.
Cf. A049347.
Programs
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PARI
\\ See Links section.
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)
Conjecture: a(n) = 2*(21*n + 3*A049347(n+2)/2)/9 for n > 4. - Stefano Spezia, Nov 24 2024
Extensions
More terms from Rémy Sigrist, Jan 21 2018
Comments