

A298035


Coordination sequence of Dual(3.12.12) tiling with respect to a trivalent node.


23



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
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OFFSET

0,2


COMMENTS

This tiling is sometimes called the triakis triangular tiling.


REFERENCES

Chaim GoodmanStrauss and N. J. A. Sloane, The Coloring BookApproach to Finding Coordination Sequences, 2018 [will be added here soon]


LINKS

Colin Barker, Table of n, a(n) for n = 0..1000
Tom Karzes, Tiling Coordination Sequences
N. J. A. Sloane, Illustration of initial terms (shows one 120degree sector of graph).
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of GrünbaumShephard 1987 with Anumbers added and in some cases the name in the RCSR database]
Index entries for linear recurrences with constant coefficients, signature (2,1).


FORMULA

Theorem: a(0)=1; thereafter a(n) = 18*n15. [Proof: Use the "coloring book" method described in the GoodmanStrauss & Sloane article.]
From Colin Barker, Jan 22 2018: (Start)
G.f.: (1 + x + 16*x^2) / (1  x)^2.
a(n) = 2*a(n1)  a(n2) for n>2.
(End)


MAPLE

f3:=proc(n) if n=0 then 1 else 18*n15; fi; end;
[seq(f3(n), n=0..80)];


PROG

(PARI) Vec((1 + x + 16*x^2) / (1  x)^2 + O(x^60)) \\ Colin Barker, Jan 22 2018


CROSSREFS

Cf. A019557 (12valent node), A016790 (partial sums, provided its offset is changed).
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: A191763 A227241 A076169 * A178082 A056387 A056377
Adjacent sequences: A298032 A298033 A298034 * A298036 A298037 A298038


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Jan 22 2018


STATUS

approved



