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Coordination sequence for the cpq net with respect to a node where a square, hexagon, and octagon meet.
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%I #13 Feb 28 2024 05:54:40

%S 1,3,5,8,12,15,16,18,23,27,28,29,33,38,40,41,44,48,51,53,56,59,61,64,

%T 68,71,72,74,79,83,84,85,89,94,96,97,100,104,107,109,112,115,117,120,

%U 124,127,128,130,135,139,140,141,145,150,152,153,156,160,163,165,168,171,173,176

%N Coordination sequence for the cpq net with respect to a node where a square, hexagon, and octagon meet.

%C 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.

%H Chaim Goodman-Strauss and N. J. A. Sloane, <a href="https://doi.org/10.1107/S2053273318014481">A Coloring Book Approach to Finding Coordination Sequences</a>, Acta Cryst. A75 (2019), 121-134, also <a href="http://NeilSloane.com/doc/Cairo_final.pdf">on NJAS's home page</a>. Also <a href="http://arxiv.org/abs/1803.08530">on arXiv</a>, arXiv:1803.08530 [math.CO], 2018-2019.

%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/cpq">The cpq tiling (or net)</a>

%H N. J. A. Sloane, <a href="/A348227/a348227_8.pdf">A portion of the cpq net</a> (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).

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,0,1,-1,1,-1).

%F 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.)

%t 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 *)

%Y Cf. A348227-A348239.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Oct 10 2021