A245540 -id:A245540 - cp's OEIS frontend

A245180 A160239(n)/8.

1, 1, 3, 1, 8, 3, 14, 1, 8, 8, 24, 3, 24, 14, 52, 1, 8, 8, 24, 8, 64, 24, 112, 3, 24, 24, 72, 14, 112, 52, 216, 1, 8, 8, 24, 8, 64, 24, 112, 8, 64, 64, 192, 24, 192, 112, 416, 3, 24, 24, 72, 24, 192, 72, 336, 14, 112, 112, 336, 52, 416, 216, 848, 1, 8, 8, 24, 8, 64, 24, 112, 8, 64, 64, 192

Offset: 1

N. J. A. Sloane, Jul 16 2014

			The entries may be arranged into blocks of sizes 1,2,4,8,...:
B_0: 1,
B_1: 1, 3,
B_2: 1, 8, 3, 14,
B_3: 1, 8, 8, 24, 3, 24, 14, 52,
B_4: 1, 8, 8, 24, 8, 64, 24, 112, 3, 24, 24, 72, 14, 112, 52, 216,
B_5: 1, 8, 8, 24, 8, 64, 24, 112, 8, 64, 64, 192, 24, 192, 112, 416, 3, 24, 24, 72, 24, 192, 72, 336, 14, 112, 112, 336, 52, 416, 216, 848,
...
The first half of each block is equal to 1 followed by 8 times an initial segment of the sequence itself.
The next quarter of each block consists of 3 times (1 followed by 8 times an initial segment of the sequence itself).
The next one-eighth of each block consists of 14 times (1 followed by 8 times an initial segment of the sequence itself).
And so on, the successive multipliers 1,3,14,52,... being given by A083424.
Also, the final quarter of any block consists of the twice the last half of the previous block added to eight times the full block before that.
Consider for example the 4th block,
[1, 8, 8, 24, 8, 64, 24, 112; 3, 24, 24, 72; 14, 112, 52, 216].
This is [1 8*(1,1,3,1,8,3,14); 3*(1 8*(1,1,3)); 2*(3,24,14,52)+8*(1,8,3,14)].
The final entries in the blocks give A083424.
See also the formula section.
.
From _Omar E. Pol_, Mar 18 2015: (Start)
Also, the sequence can be written as an irregular tetrahedron T(s,r,k) as shown below:
1;
..
1;
3;
........
1,    8;
3;
14;
................
1,    8,  8, 24;
3,   24;
14;
52;
..................................
1,    8,  8, 24,  8,  64, 24, 112;
3,   24, 24, 72;
14, 112;
52;
216;
.....................................................................
1,    8,  8, 24,  8,  64, 24, 112, 8, 64, 64, 192, 24, 192, 112, 416;
3,   24, 24, 72, 24, 192, 72, 336;
14, 112,112,336;
52, 416;
216;
848;
...
Note that T(s,r,k) = T(s+1,r,k).
(End)

N. J. A. Sloane, Table of n, a(n) for n = 1..16383
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.], which is also available at arXiv:1004.3036v2

Cf. A160239, A083424, A245190, A245195, A245196, A245562, A245540 (partial sums).

See A245181 for the numbers that appear.

a245180 = flip div 8 . a160239  -- Reinhard Zumkeller, Feb 13 2015

R:=proc(n) option remember;
if n=1 then 1
elif (n mod 2) = 0 then R(n/2)
elif (n mod 4) = 3 then 2*R((n-1)/2)+R(n-2)
else 8*R((n-1)/4); fi; end;
[seq(R(n),n=1..200)];

R[n_] := R[n] = Which[n == 1, 1, Mod[n, 2] == 0, R[n/2], Mod[n, 4] == 3, 2*R[(n - 1)/2] + R[n - 2], True, 8*R[(n - 1)/4] ];
Array[R, 200] (* Jean-François Alcover, Nov 16 2017, translated from Maple *)

The following is a fairly simple explicit formula for a(n) as a function of n: a(n) = 8^(r-1) * Product_{lengths i of runs of 1 in binary expansion of n} R(i), where r is the number of runs of 1 in the binary expansion of n and R(i) = A083424(i-1) = (5*4^(i-1)+(-2)^(i-1))/6. Note that row i of the table in A245562 lists the lengths of runs of 1 in binary expansion of i. Example: n = 27 = 11011 in binary, there are two runs each of length 2, so r=1, R(2) = A083424(1) = 3, and so a(27) = 8^1*3*3 = 72. - N. J. A. Sloane, Aug 10 2014
Many 2-D cellular automata studied in the Toothpick paper (Applegate et al.) have a recursive formula for the general term in a typical block of 2^k terms (see Equations 2, 4, 5, 9, 10 12, 38, 39 of that paper). An analogous formula for the present sequence is the following.
Consider the block B_{k-1} containing terms a(2^(k-1)), a(2^(k-1)+1), ..., a(2^k-1). It is convenient to index the terms working backwards from the next, 2^k-th, term. For n in the range 2^(k-1) <= n < 2^k, write n = 2^k-2^r+j, with 0 <= r <= k-1 and 0 <= j < 2^(r-1), and j=0 if r=0. Then
(if j=0) a(2^k-2^r) = f(k-r-1),
(if j>0) a(2^k-2^r+j) = 8*f(k-r-1)*a(j),
where f(i) = A083424(i) = (5*4^i+(-2)^i)/6.
For example, here is block B_4, consisting of terms a(16)=a(31), so k=5:
n:   16 17 18 19 20 21 22 23 24 25 26 27 28  29 30 31
a(n): 1  8  8 24  8 64 24 112 3 24 24 72 14 112 52 216
r:    4  4  4  4  4  4  4  4  3  3  3  3  2   2  1  0
j:    0  1  2  3  4  5  6  7  0  1  2  3  0   1  0  0
Then we have a(24) = a(32-8) = f(5-3-1) = f(1) = 3, illustrating the first equation, and a(21) = a(32-16+5) = 8*f(0)*a(5) = 8*1*8 = 64, illustrating the second equation.
See A245196 for a list of sequences produced by this type of recurrence.

A319018 Number of ON cells after n generations of two-dimensional automaton based on knight moves (see Comments for definition).

Original entry on oeis.org

0, 1, 9, 17, 57, 65, 121, 145, 265, 273, 329, 377, 617, 657, 865, 921, 1201, 1209, 1265, 1313, 1553, 1617, 2001, 2121, 2689, 2745, 3009, 3153, 3841, 3953, 4513, 4649, 5297, 5305, 5361, 5409, 5649, 5713, 6097, 6233, 6881, 6953, 7353, 7585, 8713, 8913, 9961

Offset: 0

Rémy Sigrist, Sep 08 2018

nonn

The cells are the squares of the standard square grid.
Cells are either OFF or ON, once they are ON they stay ON forever.
Each cell has 8 neighbors, the cells that are a knight's move away.
We begin in generation 1 with a single ON cell.
A cell is turned ON at generation n+1 if it has exactly one ON neighbor at generation n.
(Since cells stay ON, an equivalent definition is that a cell is turned ON at generation n+1 if it has exactly one neighbor that has been turned ON at some earlier generation. - N. J. A. Sloane, Dec 19 2018)
This sequence has similarities with A151725: here we use knight moves, there we use king moves.
This is a knight's-move version of the Ulam-Warburton cellular automaton (see A147562). - N. J. A. Sloane, Dec 21 2018
The structure has dihedral D_8 symmetry (quarter-turn rotations plus reflections, which generate the dihedral group D_8 of order 8), so A319019 is a multiple of 8 (compare A322050). - N. J. A. Sloane, Dec 16 2018
From Omar E. Pol, Dec 16 2018: (Start)
For n >> 1 (for example: n = 257) the structure of this sequence is similar to the structure of both A194270 and of A220500, the D-toothpick cellular automata of the second kind and of the third kind respectively. The animations of both CAs are in the Applegate's movie version.
Also, the graph of A319018 is a bit similar to the graph of A245540, which is essentially a 45-degree-3D-wedge of A245542 (a pyramid) which is the partial sums of A160239 (Fredkin's replicator). See "Plot 2": A319018 vs. A245540. (End)
The conjecture that A322050(2^k+1)=1 also suggests a fractal geometry. Let P_k be the associated set of eight points. It appears that P_k may be written as the intersection of four fixed lines, y = +-2*x and x = +-2*y, with a circle, x^2 + y^2 = 5*4^k (see linked image "Log-Periodic Coloring"). - Bradley Klee, Dec 16 2018
In many of these toothpick or cellular automata sequences it is common to see graphs which look like some version of the famous blancmange curve (also known as the Takagi curve). I expect that is what we are seeing when we look at the graph of A322049, although we probably need to go a lot further out before the true shape becomes apparent. - N. J. A. Sloane, Dec 17 2018
The graph of A322049 (related to first differences of this sequence) appears to have rather a self-similar structure which repeats at powers of 2, and more specifically at 2^10 = 1024. There is no central symmetry or continuity, which are characteristic properties of the blancmange curve. - M. F. Hasler, Dec 28 2018
The 8 points added in generation n = 2^k + 1 are P_k = 2^k*K where K = {(+-2, +-1), (+-1, +-2)} is the set of the initial 8 knight moves. So P_k is indeed the intersection of the rays of slope +-1/2 resp. +-2 and a circle of radius 2^k*sqrt(5). In the subsequent generation n = 2^k + 2, the new cells switched on are exactly the 7 "new" knight move neighbors of these 8 cells, (P_k + K) \ (2^k - 1)*K. The 8th neighbor, lying one knight move closer to the origin, has been switched on in generation 2^k, together with an octagonal "wall" consisting of every other cell on horizontal and vertical segments between these points (2^k - 1)*K, and all cells on the diagonal segments between these points, as well as 2 more diagonals just next to these (on the inner side) and shorter by 2 cells (so they are empty for k = 1). This yields 4*(2 + (2^k - 2)*(1+3)) new ON cells in generation 2^k, plus 8*(2^(k-1) - 2) more new ON cells on horizontal, vertical and diagonal lines 4 units closer to the origin for k > 2, and similar additional terms for k > 4 etc. - M. F. Hasler, Dec 28 2018

Rémy Sigrist, Table of n, a(n) for n = 0..2049
David Applegate, The movie version.
Nathan Epstein, Gfycat animation of sequence.
M. F. Hasler, Interactive illustration of A319018 and A319019, Dec. 2018.
Bradley Klee, Log-Periodic Coloring at stage 257.
Bradley Klee, Log-periodic coloring of the first quadrant, over the chair tiling.
Rémy Sigrist, Illustration of the structure at stage 7
Rémy Sigrist, Illustration of the structure at stage 257
Rémy Sigrist, Colored illustration of the structure at stage 257 (where the hue is a function of the stage)
Rémy Sigrist, PARI program for A319018
N. J. A. Sloane, Hand-drawn sketch showing terms through about the eighth shell, but using offset a(0)=1. Illustrates the octagonal "castle walls".
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences

Cf. A151725, A319019 (first differences).
For further analysis see A322048, A322049, A322050, A322051.
See A322055, A322056 for a variation.
See also A139250, A147562, A194270, A220500.

A319018(n)=sum(i=1,n,A319019[i]) \\ with array A319019=A319019_upto(N) precomputed with sufficiently large N. - M. F. Hasler, Dec 28 2018

No formula or recurrence is presently known. See A322049 for a promising attack. - N. J. A. Sloane, Dec 16 2018

a(n) = Sum_{k=1..n} A319019(n) = 1 + 8*Sum_{k=2..n} A322050(n) for n >= 1. In particular, a(n) - 1 is divisible by 8 for all n >= 1. - M. F. Hasler, Dec 28 2018

Deleted an incorrect illustration. - N. J. A. Sloane, Dec 17 2018

A245542 Partial sums of A160239.

Original entry on oeis.org

1, 9, 17, 41, 49, 113, 137, 249, 257, 321, 385, 577, 601, 793, 905, 1321, 1329, 1393, 1457, 1649, 1713, 2225, 2417, 3313, 3337, 3529, 3721, 4297, 4409, 5305, 5721, 7449, 7457, 7521, 7585, 7777, 7841, 8353, 8545, 9441, 9505, 10017, 10529, 12065

Offset: 0

N. J. A. Sloane, Jul 26 2014

Also, total number of cubic ON cells after n generations in a three-dimensional cellular automaton where A160239(n) gives the number of cubic ON cells in the n-th level of the structure starting from the top. An ON cell remains ON forever. The structure looks like an irregular stepped pyramid. - Omar E. Pol, Jan 27 2015

N. J. A. Sloane, Table of n, a(n) for n = 0..16383
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Index entries for sequences related to cellular automata

Cf. A116520, A160239, A245540, A253908, A253767.

a245542 n = a245542_list !! n
a245542_list = scanl1 (+) a160239_list
-- Reinhard Zumkeller, Feb 13 2015

b[n_] := b[n] = Which[n == 1, 1, Mod[n, 2] == 0, b[n/2], Mod[n, 4] == 3, 2b[(n - 1)/2] + b[n - 2], True, 8b[(n - 1)/4]];
Join[{1}, 1 + 8 Accumulate[Array[b, 43]]] (* Jean-François Alcover, Oct 01 2018, after Omar E. Pol *)

a(n) = 1 + 8*A245540(n), n >= 1. - Omar E. Pol, Mar 07 2015

Offset changed to 0 by N. J. A. Sloane, Feb 06 2015

cp's OEIS Frontend

A245180 A160239(n)/8.

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A319018 Number of ON cells after n generations of two-dimensional automaton based on knight moves (see Comments for definition).

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A245542 Partial sums of A160239.

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