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%I #92 Aug 30 2023 15:32:23
%S 1,5,11,16,21,27,32,37,43,48,53,59,64,69,75,80,85,91,96,101,107,112,
%T 117,123,128,133,139,144,149,155,160,165,171,176,181,187,192,197,203,
%U 208,213,219,224,229,235,240,245,251,256,261,267,272,277,283,288,293,299
%N Coordination sequence for 3.3.4.3.4 Archimedean tiling.
%C 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 This is the dual tiling to the Cairo tiling (cf. A296368). - _N. J. A. Sloane_, Nov 02 2018
%C 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 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 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 First differences of A301696. - _Klaus Purath_, May 23 2020
%D 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 Joseph Myers, <a href="/A219529/b219529.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 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 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H 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 Chaim Goodman-Strauss and N. J. A. Sloane, <a href="/A219529/a219529.eps">Trunks and branches coloring</a> (taken from preceding reference)
%H 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 Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H Kival Ngaokrajang, <a href="/A219529/a219529.pdf">Illustration of initial terms</a>
%H Reticular Chemistry Structure Resource, <a href="http://rcsr.net/layers/tts">tts</a>
%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krd">The krd tiling (or net)</a>
%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krj">The krj tiling (or net)</a>
%H 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 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 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F 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 G.f.: (x+1)^4/((x^2+x+1)*(x-1)^2). - _N. J. A. Sloane_, Feb 07 2018
%F From _G. C. Greubel_, May 27 2020: (Start)
%F a(n) = (16*n - ChebyshevU(n-1, -1/2))/3 for n>0 with a(0)=1.
%F a(n) = (A008598(n) - A049347(n-1))/3 for n >0 with a(0)=1. (End)
%p A219529:= n -> `if`(n=0, 1, (16*n +1 - `mod`(n+1,3))/3);
%p seq(A219529(n), n = 0..60); # _G. C. Greubel_, May 27 2020
%t Join[{1}, LinearRecurrence[{1,0,1,-1}, {5,11,16,21}, 60]] (* _Jean-François Alcover_, Dec 13 2018 *)
%t Table[If[n==0, 1, (16*n +1 - Mod[n+1, 3])/3], {n, 0, 60}] (* _G. C. Greubel_, May 27 2020 *)
%t CoefficientList[Series[(x+1)^4/((x^2+x+1)(x-1)^2),{x,0,70}],x] (* _Harvey P. Dale_, Jul 03 2021 *)
%o (Haskell)
%o -- Very slow, could certainly be accelerated. SST stands for Snub Square Tiling.
%o setUnion [] l2 = l2
%o setUnion (a:rst) l2 = if (elem a l2) then doRest else (a:doRest)
%o where doRest = setUnion rst l2
%o setDifference [] l2 = []
%o setDifference (a:rst) l2 = if (elem a l2) then doRest else (a:doRest)
%o where doRest = setDifference rst l2
%o adjust k = (if (even k) then 1 else -1)
%o weirdAdjacent (x,y) = (x+(adjust y),y+(adjust x))
%o sstAdjacents (x,y) = [(x+1,y),(x-1,y),(x,y+1),(x,y-1),(weirdAdjacent (x,y))]
%o sstNeighbors core = foldl setUnion core (map sstAdjacents core)
%o sstGlob n core = if (n == 0) then core else (sstGlob (n-1) (sstNeighbors core))
%o sstHalo core = setDifference (sstNeighbors core) core
%o origin = [(0,0)]
%o a219529 n = length (sstHalo (sstGlob (n-1) origin))
%o -- _Allan C. Wechsler_, Nov 30 2012
%o (Sage) [1]+[(16*n+1 -(n+1)%3)/3 for n in (1..60)] # _G. C. Greubel_, May 27 2020
%Y 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 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 Cf. A296368, A301694, A301697.
%K easy,nonn
%O 0,2
%A _Allan C. Wechsler_, Nov 21 2012
%E Corrected attributions and epistemological status in Comments; provided slow Haskell code - _Allan C. Wechsler_, Nov 30 2012
%E Extended by _Joseph Myers_, Dec 04 2014
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