A219529 Coordination sequence for 3.3.4.3.4 Archimedean tiling.

1, 5, 11, 16, 21, 27, 32, 37, 43, 48, 53, 59, 64, 69, 75, 80, 85, 91, 96, 101, 107, 112, 117, 123, 128, 133, 139, 144, 149, 155, 160, 165, 171, 176, 181, 187, 192, 197, 203, 208, 213, 219, 224, 229, 235, 240, 245, 251, 256, 261, 267, 272, 277, 283, 288, 293, 299

Offset: 0

Author

Allan C. Wechsler, Nov 21 2012

Keywords

• easy
• nonn

Comments

a(n) is the number of vertices of the 3.3.4.3.4 tiling (which has three triangles and two squares, in the given cyclic order, meeting at each vertex) whose shortest path connecting them to a given origin vertex contains n edges.

This is the dual tiling to the Cairo tiling (cf. A296368). - N. J. A. Sloane, Nov 02 2018

First few terms provided by Allan C. Wechsler; Fred Lunnon and Fred Helenius gave the next few; Fred Lunnon suggested that the recurrence was a(n+3) = a(n) + 16 for n > 1. [This conjecture is true - see the CGS-NJAS link for a proof. - N. J. A. Sloane, Dec 31 2017]

Appears also to be coordination sequence for node of type V2 in "krd" 2-D tiling (or net). This should be easy to prove by the coloring book method (see link). - N. J. A. Sloane, Mar 25 2018

Appears also to be coordination sequence for node of type V1 in "krj" 2-D tiling (or net). This also should be easy to prove by the coloring book method (see link). - N. J. A. Sloane, Mar 26 2018

First differences of A301696. - Klaus Purath, May 23 2020

References

• Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, 1st row, 2nd tiling, also 2nd row, third tiling.

Links

• Joseph Myers, Table of n, a(n) for n = 0..1000
• Giedrius Alkauskas, Colouring tiles in an isohedral tiling: automaton, defects and grain boundaries, arXiv:2301.10975 [math.CO], 2023.
• Brian Galebach, Collection of n-Uniform Tilings. See Numbers 14 and 17 from the list of 20 2-uniform tilings.
• 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 on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.
• Chaim Goodman-Strauss and N. J. A. Sloane, Trunks and branches coloring (taken from preceding reference)
• Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
• Tom Karzes, Tiling Coordination Sequences
• Kival Ngaokrajang, Illustration of initial terms
• Reticular Chemistry Structure Resource, tts
• Reticular Chemistry Structure Resource (RCSR), The krd tiling (or net)
• Reticular Chemistry Structure Resource (RCSR), The krj 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, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
• 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)
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

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.

Cf. A296368, A301694, A301697.

Programs

Haskell

-- Very slow, could certainly be accelerated.  SST stands for Snub Square Tiling.
setUnion [] l2 = l2
setUnion (a:rst) l2 = if (elem a l2) then doRest else (a:doRest)
  where doRest = setUnion rst l2
setDifference [] l2 = []
setDifference (a:rst) l2 = if (elem a l2) then doRest else (a:doRest)
  where doRest = setDifference rst l2
adjust k = (if (even k) then 1 else -1)
weirdAdjacent (x,y) = (x+(adjust y),y+(adjust x))
sstAdjacents (x,y) = [(x+1,y),(x-1,y),(x,y+1),(x,y-1),(weirdAdjacent (x,y))]
sstNeighbors core = foldl setUnion core (map sstAdjacents core)
sstGlob n core = if (n == 0) then core else (sstGlob (n-1) (sstNeighbors core))
sstHalo core = setDifference (sstNeighbors core) core
origin = [(0,0)]
a219529 n = length (sstHalo (sstGlob (n-1) origin))
-- Allan C. Wechsler, Nov 30 2012

Maple

A219529:= n -> `if`(n=0, 1, (16*n +1 - `mod`(n+1,3))/3);
seq(A219529(n), n = 0..60); # G. C. Greubel, May 27 2020

Mathematica

Join[{1}, LinearRecurrence[{1,0,1,-1}, {5,11,16,21}, 60]] (* Jean-François Alcover, Dec 13 2018 *)
Table[If[n==0, 1, (16*n +1 - Mod[n+1, 3])/3], {n, 0, 60}] (* G. C. Greubel, May 27 2020 *)
CoefficientList[Series[(x+1)^4/((x^2+x+1)(x-1)^2),{x,0,70}],x] (* Harvey P. Dale, Jul 03 2021 *)

Sage

[1]+[(16*n+1 -(n+1)%3)/3 for n in (1..60)] # G. C. Greubel, May 27 2020

Formula

Conjectured to be a(n) = floor((16n+1)/3) for n>0; a(0) = 1; this is a consequence of the suggested recurrence due to Lunnon (see comments). [This conjecture is true - see the CGS-NJAS link in A296368 for a proof. - N. J. A. Sloane, Dec 31 2017]

G.f.: (x+1)^4/((x^2+x+1)*(x-1)^2). - N. J. A. Sloane, Feb 07 2018

From G. C. Greubel, May 27 2020: (Start)

a(n) = (16*n - ChebyshevU(n-1, -1/2))/3 for n>0 with a(0)=1.

a(n) = (A008598(n) - A049347(n-1))/3 for n >0 with a(0)=1. (End)

Extensions

Corrected attributions and epistemological status in Comments; provided slow Haskell code - Allan C. Wechsler, Nov 30 2012

Extended by Joseph Myers, Dec 04 2014