

A265035


Coordination sequence of 2uniform tiling {3.4.6.4, 4.6.12} with respect to a point of type 4.6.12.


37



1, 3, 6, 9, 11, 14, 17, 21, 25, 28, 30, 32, 35, 39, 43, 46, 48, 50, 53, 57, 61, 64, 66, 68, 71, 75, 79, 82, 84, 86, 89, 93, 97, 100, 102, 104, 107, 111, 115, 118, 120, 122, 125, 129, 133, 136, 138, 140, 143, 147, 151, 154, 156, 158, 161, 165, 169, 172, 174, 176
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OFFSET

0,2


COMMENTS

Joseph Myers (Dec 14 2015) reports that "My program for coordination sequences requires describing the tiling structure under translation, listing all edges in the form: (class1, 0, 0) has an edge to (class2, x, y). The present tiling has 18 orbits of vertices under translation and 30 orbits of edges under translation (each of which is described in both directions). So in principle it could generate the other 19 2uniform tilings, but without a cross check with handcomputed terms there's a risk of e.g. missing some edges, and a fair amount of work producing all the descriptions of translation classes of edges."
Linear recurrence and g.f. confirmed by Shutov/Maleev link.  Ray Chandler, Aug 31 2023


REFERENCES

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See page 67, 4th row, 3rd tiling.
Otto Krötenheerdt, Die homogenen Mosaike nter Ordnung in der euklidischen Ebene, I, II, III, Wiss. Z. MartinLutherUniv. HalleWittenberg, MathNatur. Reihe, 18 (1969), 273290; 19 (1970), 1938 and 97122. [Includes classification of 2uniform tilings]
Anton Shutov and Andrey Maleev, Coordination sequences of 2uniform graphs, Z. Kristallogr., 235 (2020), 157166.


LINKS

N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)


FORMULA

Based on the bfile, the g.f. appears to be (1+x^2+2*x^52*x^6+2*x^7x^8)/(13*x+4*x^23*x^3+x^4). This matches the first 1000 terms, so is probably correct.  N. J. A. Sloane, Dec 14 2015
Conjectured g.f. is equivalent to a(n) = 3*n  A010892(n+1) for n >= 5.  R. J. Mathar, Oct 09 2020


MATHEMATICA

LinearRecurrence[{3, 4, 3, 1}, {1, 3, 6, 9, 11, 14, 17, 21, 25}, 100] (* Paolo Xausa, Nov 15 2023 *)


CROSSREFS

See A265036 for the other type of point.
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.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



