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A219529 Coordination sequence for 3.3.4.3.4 Archimedean tiling.

%I A219529 #92 Aug 30 2023 15:32:23
%S A219529 1,5,11,16,21,27,32,37,43,48,53,59,64,69,75,80,85,91,96,101,107,112,
%T A219529 117,123,128,133,139,144,149,155,160,165,171,176,181,187,192,197,203,
%U A219529 208,213,219,224,229,235,240,245,251,256,261,267,272,277,283,288,293,299
%N A219529 Coordination sequence for 3.3.4.3.4 Archimedean tiling.
%C A219529 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.
%C A219529 This is the dual tiling to the Cairo tiling (cf. A296368). - _N. J. A. Sloane_, Nov 02 2018
%C A219529 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]
%C A219529 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
%C A219529 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
%C A219529 First differences of A301696. - _Klaus Purath_, May 23 2020
%D A219529 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.
%H A219529 Joseph Myers, <a href="/A219529/b219529.txt">Table of n, a(n) for n = 0..1000</a>
%H A219529 Giedrius Alkauskas, <a href="https://arxiv.org/abs/2301.10975">Colouring tiles in an isohedral tiling: automaton, defects and grain boundaries</a>, arXiv:2301.10975 [math.CO], 2023.
%H A219529 Brian Galebach, <a href="http://probabilitysports.com/tilings.html">Collection of n-Uniform Tilings</a>. See Numbers 14 and 17 from the list of 20 2-uniform tilings.
%H A219529 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H A219529 Chaim Goodman-Strauss and N. J. A. Sloane, <a href="https://doi.org/10.1107/S2053273318014481">A Coloring Book Approach to Finding Coordination Sequences</a>, Acta Cryst. A75 (2019), 121-134, also <a href="http://NeilSloane.com/doc/Cairo_final.pdf">on NJAS's home page</a>. Also on <a href="http://arxiv.org/abs/1803.08530">arXiv</a>, arXiv:1803.08530 [math.CO], 2018-2019.
%H A219529 Chaim Goodman-Strauss and N. J. A. Sloane, <a href="/A219529/a219529.eps">Trunks and branches coloring</a> (taken from preceding reference)
%H A219529 Branko Grünbaum and Geoffrey C. Shephard, <a href="http://www.jstor.org/stable/2689529">Tilings by regular polygons</a>, Mathematics Magazine, 50 (1977), 227-247.
%H A219529 Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H A219529 Kival Ngaokrajang, <a href="/A219529/a219529.pdf">Illustration of initial terms</a>
%H A219529 Reticular Chemistry Structure Resource, <a href="http://rcsr.net/layers/tts">tts</a>
%H A219529 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krd">The krd tiling (or net)</a>
%H A219529 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krj">The krj tiling (or net)</a>
%H A219529 Anton Shutov and Andrey Maleev, <a href="https://doi.org/10.1515/zkri-2020-0002">Coordination sequences of 2-uniform graphs</a>, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
%H A219529 N. J. A. Sloane, <a href="/A008576/a008576.png">The uniform planar nets and their A-numbers</a> [Annotated scanned figure from Gruenbaum and Shephard (1977)]
%H A219529 N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, <a href="https://vimeo.com/314786942">Part I</a>, <a href="https://vimeo.com/314790822">Part 2</a>, <a href="https://oeis.org/A320487/a320487.pdf">Slides.</a> (Mentions this sequence)
%H A219529 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F A219529 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]
%F A219529 G.f.: (x+1)^4/((x^2+x+1)*(x-1)^2). - _N. J. A. Sloane_, Feb 07 2018
%F A219529 From _G. C. Greubel_, May 27 2020: (Start)
%F A219529 a(n) = (16*n - ChebyshevU(n-1, -1/2))/3 for n>0 with a(0)=1.
%F A219529 a(n) = (A008598(n) - A049347(n-1))/3 for n >0 with a(0)=1. (End)
%p A219529 A219529:= n -> `if`(n=0, 1, (16*n +1 - `mod`(n+1,3))/3);
%p A219529 seq(A219529(n), n = 0..60); # _G. C. Greubel_, May 27 2020
%t A219529 Join[{1}, LinearRecurrence[{1,0,1,-1}, {5,11,16,21}, 60]] (* _Jean-François Alcover_, Dec 13 2018 *)
%t A219529 Table[If[n==0, 1, (16*n +1 - Mod[n+1, 3])/3], {n, 0, 60}] (* _G. C. Greubel_, May 27 2020 *)
%t A219529 CoefficientList[Series[(x+1)^4/((x^2+x+1)(x-1)^2),{x,0,70}],x] (* _Harvey P. Dale_, Jul 03 2021 *)
%o A219529 (Haskell)
%o A219529 -- Very slow, could certainly be accelerated.  SST stands for Snub Square Tiling.
%o A219529 setUnion [] l2 = l2
%o A219529 setUnion (a:rst) l2 = if (elem a l2) then doRest else (a:doRest)
%o A219529   where doRest = setUnion rst l2
%o A219529 setDifference [] l2 = []
%o A219529 setDifference (a:rst) l2 = if (elem a l2) then doRest else (a:doRest)
%o A219529   where doRest = setDifference rst l2
%o A219529 adjust k = (if (even k) then 1 else -1)
%o A219529 weirdAdjacent (x,y) = (x+(adjust y),y+(adjust x))
%o A219529 sstAdjacents (x,y) = [(x+1,y),(x-1,y),(x,y+1),(x,y-1),(weirdAdjacent (x,y))]
%o A219529 sstNeighbors core = foldl setUnion core (map sstAdjacents core)
%o A219529 sstGlob n core = if (n == 0) then core else (sstGlob (n-1) (sstNeighbors core))
%o A219529 sstHalo core = setDifference (sstNeighbors core) core
%o A219529 origin = [(0,0)]
%o A219529 a219529 n = length (sstHalo (sstGlob (n-1) origin))
%o A219529 -- _Allan C. Wechsler_, Nov 30 2012
%o A219529 (Sage) [1]+[(16*n+1 -(n+1)%3)/3 for n in (1..60)] # _G. C. Greubel_, May 27 2020
%Y A219529 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).
%Y A219529 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.
%Y A219529 Cf. A296368, A301694, A301697.
%K A219529 easy,nonn
%O A219529 0,2
%A A219529 _Allan C. Wechsler_, Nov 21 2012
%E A219529 Corrected attributions and epistemological status in Comments; provided slow Haskell code - _Allan C. Wechsler_, Nov 30 2012
%E A219529 Extended by _Joseph Myers_, Dec 04 2014