A235643 -id:A235643 - cp's OEIS frontend

A274728 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-1,-2) (-2,-1) or (-1,0) and new values introduced in order 0..2.

1, 2, 1, 5, 6, 2, 14, 24, 16, 4, 41, 96, 68, 48, 8, 122, 384, 296, 260, 144, 16, 365, 1536, 1300, 1632, 1040, 432, 32, 1094, 6144, 5728, 10368, 9308, 4132, 1296, 64, 3281, 24576, 25268, 66132, 84948, 52912, 16524, 3888, 128, 9842, 98304, 111512, 421904, 771300

Offset: 1

R. H. Hardin, Jul 03 2016

Table starts
...1.....2.......5.......14.........41..........122...........365
...1.....6......24.......96........384.........1536..........6144
...2....16......68......296.......1300.........5728.........25268
...4....48.....260.....1632......10368........66132........421904
...8...144....1040.....9308......84948.......771300.......7016284
..16...432....4132....52912.....687248......8960464.....116633012
..32..1296...16524...301784....5634776....105328812....1980677448
..64..3888...66088..1721528...45990340...1237030912...33415073716
.128.11664..264332..9816048..375845052..14505195904..564905405336
.256.34992.1057316.55981988.3070055916.170291855288.9541460598408

			Some solutions for n=4 k=4
..0..1..1..2. .0..1..2..0. .0..1..2..2. .0..1..2..1. .0..1..2..1
..1..0..2..0. .1..0..1..2. .1..0..1..0. .1..0..1..2. .1..0..1..0
..0..2..0..2. .0..1..0..1. .2..1..0..1. .2..1..0..1. .0..1..0..1
..2..0..2..1. .1..2..2..2. .1..2..1..2. .1..2..1..0. .1..2..1..2

R. H. Hardin, Table of n, a(n) for n = 1..220

Column 1 is A000079(n-2).
Row 1 is A007051(n-1).
Row 2 is A002023(n-2).
Row 3 is A235643(n-1).

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>2
k=2: a(n) = 3*a(n-1) for n>3
k=3: a(n) = 3*a(n-1) +4*a(n-2) +a(n-3) -3*a(n-4) -4*a(n-5) for n>7
k=4: [order 8] for n>12
k=5: [order 15] for n>19
k=6: [order 30] for n>35
k=7: [order 59] for n>65
Empirical for row n:
n=1: a(n) = 4*a(n-1) -3*a(n-2)
n=2: a(n) = 4*a(n-1) for n>2
n=3: a(n) = 6*a(n-1) -7*a(n-2) for n>3
n=4: a(n) = 8*a(n-1) -10*a(n-2) -4*a(n-3) +13*a(n-4) -7*a(n-5) +a(n-6) for n>7
n=5: [order 7] for n>9
n=6: [order 18] for n>20
n=7: [order 30] for n>34

A235648 Perimeter (rounded down) of a tetraflake-like fractal after n iterations, a(1) = 1 (see comments).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 16, 25, 39, 61, 97, 155, 249, 404, 657, 1073, 1759, 2892, 4768, 7877, 13036, 21602, 35838, 59508, 98885, 164416, 273502, 455137, 757628, 1261470, 2100791, 3499106, 5828894, 9710891, 16179575, 26958966, 44922289, 74858052, 124746848, 207889317

Offset: 1

Kival Ngaokrajang, Apr 20 2014

nonn

Construction rule is same as for box and Vicsek fractals, but uses 6 boxes at initial stage (n = 1) and has only one symmetrical axis. The scale factor of these fractals is 1/3. The actual tetraflake fractals have a scale factor of 1/2.

The total number of sides at different lengths of a tetraflake-like fractal after n iterations is A235643(n). The total number of holes is A241271(n+1).

Kival Ngaokrajang, Illustration of initial terms
Eric Weisstein's World of Mathematics, Box Fractal
Wikipedia, n-flake
Wikipedia, Vicsek Fractal

Cf. A241271, A235643.
Cf. A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240840 (hendecaflake), A240735 (dodecaflake), A240841 (tridecaflake).
Cf. A063628 (hexaflake).
Cf. A240916, A240917 (triflake-like); A238777 (tetraflake-like).

{a=18;c=1;print1(1,", "); for (n=1,50, c=4*c+3^(n-1); a=5*a-2*c; aa=floor((a*(1/3)^n)/18); print1(aa,", "));}

Floor((5*a(n-1)-2*(4*c(n-1)+3^(n-1)))/18) for n >1, a(1)=18, c(1)=1.

cp's OEIS Frontend

A274728 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-1,-2) (-2,-1) or (-1,0) and new values introduced in order 0..2.

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