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A348236
Coordination sequence for the cpq net with respect to a node where a square, hexagon, and octagon meet.
2
1, 3, 5, 8, 12, 15, 16, 18, 23, 27, 28, 29, 33, 38, 40, 41, 44, 48, 51, 53, 56, 59, 61, 64, 68, 71, 72, 74, 79, 83, 84, 85, 89, 94, 96, 97, 100, 104, 107, 109, 112, 115, 117, 120, 124, 127, 128, 130, 135, 139, 140, 141, 145, 150, 152, 153, 156, 160, 163, 165, 168, 171, 173, 176
OFFSET
0,2
COMMENTS
The cpq net is the dual graph to the 123-circle graph G studied in A348227-A348235. Thanks to Davide M. Proserpio for pointing this out.
LINKS
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.
Reticular Chemistry Structure Resource (RCSR), The cpq tiling (or net)
N. J. A. Sloane, A portion of the cpq net (the numbers correspond to the coordination sequence for nodes in the first quadrant, with respect to a base point in the lower left corner of the picture).
FORMULA
G.f. = (1+q)^2*(q^6+2*q^4+q^3+2*q^2+1) / ((1-q)*(1+q^2)*(1-q^5)). (Discovered and proved using the "coloring book" method.)
MATHEMATICA
CoefficientList[Series[(1+x)^2(x^6+2x^4+x^3+2x^2+1)/((1-x)(1+x^2)(1-x^5)), {x, 0, 100}], x] (* or *) LinearRecurrence[{1, -1, 1, 0, 1, -1, 1, -1}, {1, 3, 5, 8, 12, 15, 16, 18, 23}, 100] (* Harvey P. Dale, Jan 13 2023 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 10 2021
STATUS
approved