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A298022
Coordination sequence for Dual(3^3.4^2) tiling with respect to a trivalent node.
22
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
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).
Frank Morgan, Optimal Pentagonal Tilings, Video, May 2021. [Mentions this tiling]
Reticular Chemistry Structure Resource (RCSR), The cem-d tiling (or net)
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]
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
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.
Cf. A049347.
Sequence in context: A332263 A376998 A310248 * A227133 A170883 A198463
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 21 2018
EXTENSIONS
More terms from Rémy Sigrist, Jan 21 2018
STATUS
approved