A319018 -id:A319018 - cp's OEIS frontend

A319019 First differences of A319018.

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

A322662 a(n) is to A151723(n+1) as A319018(n+1) is to A147562(n+1), n >= 0.

Original entry on oeis.org

1, 13, 25, 109, 121, 193, 325, 493, 529, 661, 829, 1129, 1189, 1405, 1657, 2101, 2149, 2281, 2533, 3133, 3337, 3709, 4309, 4909, 5065, 5449, 5917, 6757, 6877, 7381, 7873, 8845, 8893, 9025, 9277, 9877, 10165, 10849, 11737

Offset: 0

Bradley Klee, Dec 22 2018

nonn

Also the number of ON cells after n generations in a knight's-move, one-neighbor, accumulative cellular automaton on the hexagonal lattice A_2. Define v(m)=2*sqrt(3)*[cos(m*Pi/3+Pi/6), sin(m*Pi/3+Pi/6)], vL(m)=2*v(m)+v(m+1), vR(m)=2*v(m)+v(m-1). The set of "knight's moves", M={vL(m):m=1,2,..6} U {vR(m):m=1,2,..6}, follows from an analogy between Z^2 and A_2. At each generation all ON cells remain ON while an OFF cell turns ON if and only if it has exactly one M-neighbor in the previous generation.
Fractal Structure Theorem (FST). A pair of lattice vectors M={v1,v2} generate a wedge, W = {x*v1 + y*v2 : x>=0, y>=0}. Define W-Subsets T_k such that T_{k+1}= T_k U { 2^n*v1 + v : v in T_k } U {2^n*v2 + v : v in T_k}, T_0 = { [0,0] }. The limit set T_{oo} is a fractal, and acquires the topology of a binary tree when points are connected by either v1 or v2. As a tree, T_k has height 2^k-1, with 2^k vertices at maximum depth, along a line in the direction v1-v2. Assume a one-M-neighbor, accumulative cellular automaton on W, where all vertices in T_k are ON. In the next generation, the front F_k={2^k*v1+m*(v2-v1) : 0<=m<=2^k} contains only two ON cells, {2^k*v1,2^k*v2}. The spacing, 2^k-1, is wide enough to turn ON two copies of T_k, one starting from each of the two ON cells in F_k. Thus T_{k+1} is also ON. Whenever only T_0 is ON as an initial condition, by induction, T_{oo} is ultimately ON.
The FST applies here to 12 distinct wedges: with {v1,v2}={vL(m), vR(m)} or with (v1,v2)={vL(m), vR(m+1)}, and m=1,2,..6. The triangle inequality ensures that paths including other vectors cannot reach the front F_k by generation 2^k. However, other vectors do generate retrogressive growth, which turns ON many additional cells.
The FST applies to a wide range of Cellular Automata. Wolfram's one-dimensional rule 90 gives the most elementary example where T_{oo} determines every ON cell. The tree structure T_{oo} also occurs with two-dimensional, accumulative, one-neighbor C.A. such as A151723, A319018, A147562. Also try: M={[0,1],[0,-1],[2,1],[-2,-1]}.
According to S. Ulam (cf. Links), some version of the FST was already known to J. Holladay circa 1960.
The FST implies scale resonance between this cellular automaton and the arrowed half hexagon tiling (cf. Links).

Bradley Klee, Log-Periodic Coloring over Arrowed Half Hexagon tiling.
Bradley Klee, Log-Periodic Coloring to Stage 64.
Bradley Klee, T_n Tree Structure, n=1,2,3,4.
Bradley Klee, Limit-Periodic Tilings, Wolfram Demonstrations Project (2015).
S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 216 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 [Annotated scanned copy]

Hexagonal: A151723. Square: A319018, A147562. Tree: A006046, A267700, A038573. A322663.

HexStar=2*Sqrt[3]*{Cos[#*Pi/3+Pi/6],Sin[#*Pi/3+Pi/6]}&/@Range[0,5];
MoveSet=Join[2*HexStar+RotateRight[HexStar],2*HexStar+RotateLeft[HexStar]];
Clear@Pts;Pts[0] = {{0, 0}};
Pts[n_]:=Pts[n]=With[{pts=Pts[n-1]},Union[pts,Cases[Tally[Flatten[pts/.{x_,y_}:> Evaluate[{x,y}+#&/@MoveSet],1]],{x_,1}:>x]]];Length[Pts[#]]&/@Range[0,32]

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.

A322048 Final elements in rows when A322050 is displayed as a triangle.

Original entry on oeis.org

1, 5, 15, 35, 81, 173, 357, 725, 1461, 2933, 5877, 11765, 23541, 47093, 94197, 188405

Offset: 1

N. J. A. Sloane, Dec 15 2018

Needs more terms and a b-file.

Cf. A319018, A319019, A322049, A322050.

Conjectures: a(n) = 2*a(n-1) + 11 for n >= 5; G.f. = x*(6*x^4+2*x^2+2*x+1)/((1-x)*(1-2*x)).

a(n) = A322050(2^n) = A319019(2^n)/8. - M. F. Hasler, Dec 27 2018

a(12)-a(16) from Rémy Sigrist, Dec 17 2018

Offset changed from 0 to 1 by M. F. Hasler, Dec 27 2018

A322055 Number of ON cells after n generations of two-dimensional automaton based on knight moves (see Comments for definition; here a cell is turned ON if 1 or 2 neighbors are ON).

Original entry on oeis.org

1, 9, 41, 73, 145, 185, 321, 385, 577, 649, 881, 993, 1297, 1401, 1729, 1889, 2305, 2441, 2865, 3073, 3601, 3769, 4289, 4545, 5185, 5385, 6001, 6305, 7057, 7289, 8001, 8353, 9217, 9481, 10289, 10689, 11665, 11961, 12865, 13313, 14401, 14729, 15729, 16225

Offset: 0

N. J. A. Sloane, Dec 21 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 0 with a single ON cell.
A cell is turned ON at generation n+1 if it has either one or two 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 one or two neighbors that has been turned ON at some earlier generation.
This sequence is a variant of A319018.
This is another knight's-move version of the Ulam-Warburton cellular automaton (see A147562).
The structure has dihedral D_8 symmetry (quarter-turn rotations plus reflections), so A322055 is a multiple of 8.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Rémy Sigrist, Illustration of the structure at stage 255
N. J. A. Sloane, Illustration of a(0) to a(5).

Cf. A139250, A147562, A319018, A319019, A322056.

Conjectures from Colin Barker, Dec 22 2018: (Start)
G.f.: (1 + 8*x + 32*x^2 + 32*x^3 + 70*x^4 + 24*x^5 + 72*x^6 + 49*x^8 - 8*x^10 + 16*x^11 - 8*x^12) / ((1 - x)^3*(1 + x)^2*(1 + x^2)^2).
a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n>8.
(End)

More terms from Rémy Sigrist, Dec 22 2018

A322056 First differences of A322055.

Original entry on oeis.org

1, 8, 32, 32, 72, 40, 136, 64, 192, 72, 232, 112, 304, 104, 328, 160, 416, 136, 424, 208, 528, 168, 520, 256, 640, 200, 616, 304, 752, 232, 712, 352, 864, 264, 808, 400, 976, 296, 904, 448, 1088, 328, 1000, 496, 1200, 360, 1096, 544, 1312, 392, 1192, 592, 1424

Offset: 0

N. J. A. Sloane, Dec 21 2018

nonn

Number of cells turned ON at generation n of the knight's-move cellular automaton described in A322055.

This is another knight's-move version of the Ulam-Warburton cellular automaton (see A147562).

Rémy Sigrist, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Illustration of a(0) to a(5).

Cf. A139250, A147562, A319018, A319019, A322055.

Conjectures from Colin Barker, Dec 22 2018: (Start)
G.f.: (1 + 8*x + 32*x^2 + 32*x^3 + 70*x^4 + 24*x^5 + 72*x^6 + 49*x^8 - 8*x^10 + 16*x^11 - 8*x^12) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2).
a(n) = 2*a(n-4) - a(n-8) for n>8.
(End)

More terms from Rémy Sigrist, Dec 22 2018

A322051 a(n) is the number of initial terms in the row of length 2^n of A322050 that agree with the limiting sequence A322049.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 22, 43, 86, 171, 342, 683, 1366, 2731, 5462

Offset: 0

Hugo Pfoertner, Dec 16 2018

nonn

Seems to be identical to A005578 with the exception of a(3) = 4. - Omar E. Pol, Dec 17 2018

			   n     i*    a(n)  first non-matching pair    (i* = Index of start in A319018)
   0      3     1      5      1
   1      5     1      7      5
   2      9     2      6      3
   3     17     4      8      5
   4     33     6     17     15
   5     65    11    145    141
   6    129    22     73     69
   7    257    43    734    726
   8    513    86    349    341
   9   1025   171   3579   3563
  10   2049   342   1696   1680
  11   4097   683  17810  17778
  12   8193  1366   8394   8362
  13  16385  2731  88553  88489
  14  32769  5462  41665  41601
  ...

Cf. A319018, A319019, A322049, A322050.

A334840 a(1) = 1, a(n) = a(n-1)/gcd(a(n-1),n) if this gcd is > 1, else a(n) = 4*a(n-1).

Original entry on oeis.org

1, 4, 16, 4, 16, 8, 32, 4, 16, 8, 32, 8, 32, 16, 64, 4, 16, 8, 32, 8, 32, 16, 64, 8, 32, 16, 64, 16, 64, 32, 128, 4, 16, 8, 32, 8, 32, 16, 64, 8, 32, 16, 64, 16, 64, 32, 128, 8, 32, 16, 64, 16, 64, 32, 128, 16, 64, 32, 128, 32, 128, 64, 256, 4, 16

Offset: 1

Ctibor O. Zizka, May 13 2020

nonn

A variant of A133058. - Ctibor O. Zizka, Apr 14 2023

			a(2) = 4*a(1) = 4, a(3) = 4*a(2) = 16, a(4) = a(3)/4 = 4, a(5) = 4*a(4) = 16, ...

Cf. A000120, A001316, A133058, A255140, A257806, A319018.

a:=[1]; for n in [2..70] do if Gcd(a[n-1], n) eq 1 then Append(~a,4* a[n-1]); else Append(~a, a[n-1] div Gcd(a[n-1], n)); end if; end for; a; // Marius A. Burtea, May 13 2020

a[1] = 1; a[n_] := a[n] = If[(g = GCD[a[n-1], n]) > 1, a[n-1]/g, 4*a[n-1]]; Array[a, 100] (* Amiram Eldar, May 13 2020 *)

lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, my(g = gcd(va[n-1], n)); if (g > 1, va[n] = va[n-1]/g, va[n] = 4*va[n-1]);); va;} \\ Michel Marcus, May 17 2020

from itertools import count, islice
from math import gcd
def A334840_gen(): # generator of terms
    yield (a:=1)
    for n in count(2):
        yield (a:=a<<2 if (b:=gcd(a,n)) == 1 else a//b)
A334840_list = list(islice(A334840_gen(),30)) # Chai Wah Wu, Mar 18 2023

a(n) = 2^((n mod 2) + A000120(n) + 1), for n >= 2. - Ctibor O. Zizka, Apr 15 2023

a(n) = 2*A001316(n)*(n mod 2 + 1), for n >= 2. - Ctibor O. Zizka, Apr 15 2023

cp's OEIS Frontend

A319019 First differences of A319018.

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A322662 a(n) is to A151723(n+1) as A319018(n+1) is to A147562(n+1), n >= 0.

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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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A322048 Final elements in rows when A322050 is displayed as a triangle.

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A322055 Number of ON cells after n generations of two-dimensional automaton based on knight moves (see Comments for definition; here a cell is turned ON if 1 or 2 neighbors are ON).

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A322056 First differences of A322055.

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A322051 a(n) is the number of initial terms in the row of length 2^n of A322050 that agree with the limiting sequence A322049.

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A334840 a(1) = 1, a(n) = a(n-1)/gcd(a(n-1),n) if this gcd is > 1, else a(n) = 4*a(n-1).

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