

A301289


Coordination sequence for a tetravalent node of type 3.4.3.12 in "cph" 2D tiling (or net).


38



1, 4, 5, 6, 12, 14, 15, 18, 21, 26, 28, 26, 31, 38, 37, 38, 44, 46, 47, 50, 53, 58, 60, 58, 63, 70, 69, 70, 76, 78, 79, 82, 85, 90, 92, 90, 95, 102, 101, 102, 108, 110, 111, 114, 117, 122, 124, 122, 127, 134, 133, 134, 140, 142, 143, 146, 149, 154, 156, 154
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OFFSET

0,2


REFERENCES

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, bottom row, first tiling.


LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of nUniform Tilings. See Number 2 from the list of 20 2uniform tilings.
Brian Galebach, Enlarged illustration of tiling, suitable for coloring (taken from the web site in the previous link)
Brian Galebach, kuniform tilings (k <= 6) and their Anumbers
Chaim GoodmanStrauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121134, also on NJAS's home page. Also arXiv:1803.08530.
Reticular Chemistry Structure Resource (RCSR), The cph tiling (or net)
Rémy Sigrist, Illustration of first terms
Rémy Sigrist, PARI program for A301289
N. J. A. Sloane, Trunks and branches for determining coordination sequence, central view. Blue = trunks, red = branches, green = twigs.
N. J. A. Sloane, Trunks and branches, a different scan, truncated on right but otherwise shows quadrants I and IV in detail
N. J. A. Sloane, Page 1 of the proof of the theorem: counts for various classes of nodes [Page 1 will be added as soon as I can find a widebed scanner]
N. J. A. Sloane, Page 2 of the proof of the theorem: the corresponding generating functions


FORMULA

Theorem: G.f. = (1+2*x+4*x^3+3*x^4+6*x^64*x^7+6*x^82*x^9) / ((1x)^2*(1+x^2)*(1+x^2+x^4)).
The proof uses the coloring book method described in the GoodmanStrauss & Sloane article. The trunks and branches structure is shown in the first scan. (Not yet added.) The trunks are blue, the branches are red, and the twigs are green. There is mirror symmetry about the Yaxis, and quadrants I and II are essentially identical, as are quadrants III and IV. The counts of the various classes of nodes are given in the second scan, and the corresponding generating functions are in the third scan. Adding up the different terms gives the g.f. stated above.  N. J. A. Sloane, Apr 07 2018


MATHEMATICA

Join[{1, 4}, LinearRecurrence[{2, 3, 4, 4, 4, 3, 2, 1}, {5, 6, 12, 14, 15, 18, 21, 26}, 100]] (* JeanFrançois Alcover, Aug 05 2018 *)


PROG

(PARI) See Links section.


CROSSREFS

Cf. A301287.
Coordination sequences for the 20 2uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
Sequence in context: A072623 A006144 A047429 * A310571 A280382 A055033
Adjacent sequences: A301286 A301287 A301288 * A301290 A301291 A301292


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Mar 23 2018


EXTENSIONS

More terms from Rémy Sigrist, Mar 27 2018


STATUS

approved



