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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 (list; graph; refs; listen; history; text; internal format)
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 CGS-NJAS 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 Gr├╝nbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.

Kival Ngaokrajang, Illustration of initial terms

Reticular Chemistry Structure Resource, tts

N. J. A. Sloane, The uniform planar nets and their A-numbers [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 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

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), (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

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

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Last modified February 22 21:55 EST 2018. Contains 299469 sequences. (Running on oeis4.)