A265036 Coordination sequence of 2-uniform tiling {3.4.6.4, 4.6.12} with respect to a point of type 3.4.6.4.

1, 4, 6, 7, 10, 14, 20, 24, 24, 23, 26, 34, 42, 44, 40, 37, 42, 54, 64, 64, 56, 51, 58, 74, 86, 84, 72, 65, 74, 94, 108, 104, 88, 79, 90, 114, 130, 124, 104, 93, 106, 134, 152, 144, 120, 107, 122, 154, 174, 164, 136, 121, 138, 174, 196, 184, 152, 135, 154, 194, 218

Offset

0,2

Comments

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 n-ter Ordnung in der euklidischen Ebene, I, II, III, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math-Natur. Reihe, 18 (1969), 273-290; 19 (1970), 19-38 and 97-122. [Includes classification of 2-uniform tilings]

Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166.

Links

Joseph Myers, Table of n, a(n) for n = 0..20000

Miguel Carlos Fernández-Cabo, Artisan Procedures to Generate Uniform Tilings, International Mathematical Forum, Vol. 9, 2014, no. 23, 1109-1130. [Background information]

Brian Galebach, Collection of n-Uniform Tilings. See Number 1 from the list of 20 2-uniform tilings.

Brian Galebach, The tiling {3.4.6.4, 4.6.12}, Number 1 from list of 20 2-uniform tilings. (From the previous link)

Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers

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 arXiv:1803.08530.

Reticular Chemistry Structure Resource (RCSR), The krt tiling (or net)

Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.

N. J. A. Sloane, Illustration of initial terms of A265035 (point of type 4.6.12)

N. J. A. Sloane, Illustration of initial terms of A265036 (point of type 3.4.6.4)

Index entries for linear recurrences with constant coefficients, signature (4,-8,10,-8,4,-1).

Formula

Based on the b-file, the g.f. appears to be (-2*x^9+6*x^8-8*x^7+7*x^6-2*x^5-2*x^4+5*x^3-2*x^2+1) / (x^6-4*x^5+8*x^4-10*x^3+8*x^2-4*x+1). - N. J. A. Sloane, Dec 14 2015

Mathematica

LinearRecurrence[{4, -8, 10, -8, 4, -1}, {1, 4, 6, 7, 10, 14, 20, 24, 24, 23}, 100] (* Paolo Xausa, Nov 15 2023 *)

Crossrefs

See A265035 for the other type of point.

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706(3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120(3.3.3.3.6), A250122 (3.12.12).

Coordination sequences for the 20 2-uniform 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: A358731 A023635 A317649 * A366506 A310652 A310653

Adjacent sequences: A265033 A265034 A265035 * A265037 A265038 A265039

Keyword

nonn

Author

N. J. A. Sloane, Dec 12 2015

Extensions

Extended by Joseph Myers, Dec 13 2015

b-file extended by Joseph Myers, Dec 18 2015

Status

approved