A348236 Coordination sequence for the cpq net with respect to a node where a square, hexagon, and octagon meet.

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

Table of n, a(n) for n=0..63.

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).

Index entries for linear recurrences with constant coefficients, signature (1,-1,1,0,1,-1,1,-1).

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

Cf. A348227-A348239.

Sequence in context: A285953 A287380 A339389 * A218485 A310029 A310030

Adjacent sequences: A348233 A348234 A348235 * A348237 A348238 A348239

Keyword

nonn

Author

N. J. A. Sloane, Oct 10 2021

Status

approved