A298035 Coordination sequence of Dual(3.12.12) tiling with respect to a trivalent node.
1, 3, 21, 39, 57, 75, 93, 111, 129, 147, 165, 183, 201, 219, 237, 255, 273, 291, 309, 327, 345, 363, 381, 399, 417, 435, 453, 471, 489, 507, 525, 543, 561, 579, 597, 615, 633, 651, 669, 687, 705, 723, 741, 759, 777, 795, 813, 831, 849, 867, 885, 903, 921, 939, 957, 975, 993, 1011, 1029, 1047, 1065
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.
- Tom Karzes, Tiling Coordination Sequences
- N. J. A. Sloane, Illustration of initial terms (shows one 120-degree sector of graph).
- 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]
- Index entries for linear recurrences with constant coefficients, signature (2,-1).
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.
Programs
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Maple
f3:=proc(n) if n=0 then 1 else 18*n-15; fi; end; [seq(f3(n),n=0..80)];
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PARI
Vec((1 + x + 16*x^2) / (1 - x)^2 + O(x^60)) \\ Colin Barker, Jan 22 2018
Formula
Theorem: a(0)=1; thereafter a(n) = 18*n-15. [Proof: Use the "coloring book" method described in the Goodman-Strauss & Sloane article.]
From Colin Barker, Jan 22 2018: (Start)
G.f.: (1 + x + 16*x^2) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>2.
(End)
Comments