

A219529


Coordination sequence for 3.3.4.3.4 Archimedean tiling.


20



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
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OFFSET

0,2


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.
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 CGSNJAS link in A296368 for a proof.  N. J. A. Sloane, Dec 31 2017]


LINKS

Joseph Myers, Table of n, a(n) for n = 0..1000
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227247.
Kival Ngaokrajang, Illustration of initial terms
Reticular Chemistry Structure Resource, tts
N. J. A. Sloane, The uniform planar nets and their Anumbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]


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 CGSNJAS link in A296368 for a proof.  N. J. A. Sloane, Dec 31 2017]
G.f.: (x+1)^4/((x^2+x+1)*(x1)^2).  N. J. A. Sloane, Feb 07 2018


PROG

(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), (x1, y), (x, y+1), (x, y1), (weirdAdjacent (x, y))]
sstNeighbors core = foldl setUnion core (map sstAdjacents core)
sstGlob n core = if (n == 0) then core else (sstGlob (n1) (sstNeighbors core))
sstHalo core = setDifference (sstNeighbors core) core
origin = [(0, 0)]
a219529 n = length (sstHalo (sstGlob (n1) origin))
 Allan C. Wechsler, Nov 30 2012


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).
Sequence in context: A259067 A072557 A183989 * A278397 A172327 A272666
Adjacent sequences: A219526 A219527 A219528 * A219530 A219531 A219532


KEYWORD

easy,nonn


AUTHOR

Allan C. Wechsler, Nov 21 2012


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


STATUS

approved



