A322048 -id:A322048 - cp's OEIS frontend

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

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

A319019 First differences of A319018.

Original entry on oeis.org

0, 1, 8, 8, 40, 8, 56, 24, 120, 8, 56, 48, 240, 40, 208, 56, 280, 8, 56, 48, 240, 64, 384, 120, 568, 56, 264, 144, 688, 112, 560, 136, 648, 8, 56, 48, 240, 64, 384, 136, 648, 72, 400, 232, 1128, 200, 1048, 240, 1216, 48, 216, 160, 768, 200, 1176, 352, 1664

Offset: 0

Rémy Sigrist, Sep 09 2018

nonn

Number of cells added at n-th stage to the structure of A319018.

See A319018 for further illustrations.

M. F. Hasler, Table of n, a(n) for n = 0..16384 (first 2050 terms from Rémy Sigrist)
Nathan Epstein, Gfycat animation of sequence.
M. F. Hasler, Interactive illustration of A319018 and A319019.
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".

Cf. A319018.

For further analysis see A322048, A322049, A322050.

A319019_upto(N,S=[],K=[[t\5-2,t%5-2]|t<-digits(6888528048,25)])={ vector(N,n, #N=if(n>1, S=setunion(S,N); N=vecsort(concat([[Vecsmall(Vec(n)+k)|k<-K]|n<-N])); S=setunion(Set(vecextract(N, select(i->N[i-1]==N[i],[2..#N]))),S);setminus(Set(N),S),[[0,0]]))} \\ Increase stack size with allocatemem() for N > 86. - M. F. Hasler, Dec 27 2018

A319019(n)=sum(i=2,n,A322050(i))*8+(n>0) \\ M. F. Hasler, Dec 28 2018

Apparently, a(2^k + 1) = 8 for any k >= 0.

a(n) = 8*A322050(n) for all n > 1, see there for more formulas. - M. F. Hasler, Dec 18 2018

A322049 When A322050 is displayed as a triangle the rows converge to this sequence.

Original entry on oeis.org

1, 7, 6, 30, 8, 48, 17, 81, 9, 50, 29, 145, 27, 145, 37, 189, 8, 45, 34, 166, 45, 252, 73, 342, 37, 179, 89, 425, 74, 374, 86, 412, 8, 49, 33, 165, 46, 270, 91, 436, 50, 277, 149, 734, 122, 630, 144, 723, 38, 179, 101, 488, 130, 753, 209, 990, 90, 450, 210, 991

Offset: 0

N. J. A. Sloane, Dec 15 2018

It would be nice to have a formula or recurrence. There is certainly a lot of structure.

Indices of records of a(n)/n are (1, 3, 7, 11, 23, 27, 43, 55, 87, 91, 119, 171, 183, 343, 347, 363, 367, 375, 439, 695, 731, 887, 1367, 1371, 1391, 1399, 1451, 1463, 2743, 2923, 2927, 2935, 3511, ...). The ratio a(n)/n increases roughly by 1 at each of these. We conjecture that this ratio is unbounded. We note that the record ratios occur in "clusters" at indices twice as large as the preceding cluster: 87, 91; 171, 183; 343..375; 695..731; 1367..1463; 2743..2935; ... This is compatible with the self-similar structure of the graph of this sequence, which starts over at a(2^k) = 8 for all k >= 4. (But note also the distinctive substructure repeating with period 2^10, cf. the "logarithmic plot" link.) - M. F. Hasler, Dec 18 2018

Hugo Pfoertner, Table of n, a(n) for n = 0..5461
Hugo Pfoertner, Logarithmic plot of 5462 terms, use zoom to see details.

Cf. A319018, A319019, A322048, A322050, A322051.

From M. F. Hasler, Dec 18 2018: (Start)
Experimental data suggests the following properties:
Sporadic values occurring only a finite number of times, with no regular pattern:
  a(n) | 1 | 6 | 7 | 9 |   37   | 48 |  50   | 53 | ...
  -----+---+---+---+---+--------+----+-------+----+-----
    n  | 0 | 2 | 1 | 8 | 14, 24 |  5 | 9, 40 | 80 | ...
Values occurring in regular patterns:
a(n) = 8 iff n = 2^k, k = 2 or k >= 4; a(n) > 8 for all other n > 2.
a(n) = 33 iff n = 2^(2k+1) + 2, k >= 2; a(n) > 33 for all other n > 12 unless n = 2^k <=> a(n) = 8.
a(n) = 34 iff n = 4^k + 2, k >= 2.
a(n) = 38 iff n = 3*2^k, k = 4, 5, 6, 8, 10, ...
a(n) = 27*2^m if n = 3*2^k with k = 2 (m = 0) or k = 7, 9, ... (m = 1, 2, ...)
a(n) = 45 iff n = 20 or n = 4^k + 1, k >= 2.
a(n) = 46 iff n = 2^(2k+1) + 4, k >= 2.
a(n) = 49 iff n = 2^(2k+1) + 1, k >= 2, or n = 4^k + 4, k >= 3.
a(n) > 50 for all n > 10 not mentioned above. (End)

A322050 a(n) = A319019(n)/8.

Original entry on oeis.org

1, 1, 5, 1, 7, 3, 15, 1, 7, 6, 30, 5, 26, 7, 35, 1, 7, 6, 30, 8, 48, 15, 71, 7, 33, 18, 86, 14, 70, 17, 81, 1, 7, 6, 30, 8, 48, 17, 81, 9, 50, 29, 141, 25, 131, 30, 152, 6, 27, 20, 96, 25, 147, 44, 208, 17, 81, 42, 198, 32, 158, 37, 173, 1, 7, 6, 30, 8

Offset: 2

N. J. A. Sloane, Dec 15 2018

This is the number of new cells turned ON at the n-th generations in a 45-degree sector of the knights-move analog of the Ulam-Warburton cellular automaton defined in A319018.

			This sequence may be redrawn as an array with rows of lengths 1, 2, 4, 8, 16, ...:
1,
1, 5,
1, 7, 3, 15,
1, 7, 6, 30, 5, 26, 7, 35,
1, 7, 6, 30, 8, 48, 15, 71, 7, 33, 18, 86, 14, 70, 17, 81,
1, 7, 6, 30, 8, 48, 17, 81, 9, 50, 29, 141, 25, 131, 30, 152, 6, 27, 20, 96, 25, 147, 44, 208, 17, 81, 42, 198, 32, 158, 37, 173,
...
See A322048 for the final element of these rows; see A322049 for the sequence to which the rows converge, with additional formulas for the n-th element of each (sufficiently long) row. - _M. F. Hasler_, Dec 18 2018

Rémy Sigrist, Table of n, a(n) for n = 2..65536 (first 2048 terms from Hugo Pfoertner)
Rémy Sigrist, C# program for A322050

Cf. A319018, A319019, A322048, A322049.

cp's OEIS Frontend

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

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A319019 First differences of A319018.

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A322049 When A322050 is displayed as a triangle the rows converge to this sequence.

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A322050 a(n) = A319019(n)/8.

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