A162349 -id:A162349 - cp's OEIS frontend

A147582 First differences of A147562.

1, 4, 4, 12, 4, 12, 12, 36, 4, 12, 12, 36, 12, 36, 36, 108, 4, 12, 12, 36, 12, 36, 36, 108, 12, 36, 36, 108, 36, 108, 108, 324, 4, 12, 12, 36, 12, 36, 36, 108, 12, 36, 36, 108, 36, 108, 108, 324, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 972, 4

Offset: 1

N. J. A. Sloane, Apr 29 2009

nonn

Bisection of A323651. - Omar E. Pol, Mar 04 2019

			From _Omar E. Pol_, Jun 14 2009: (Start)
When written as a triangle:
.1;
.4;
.4,12;
.4,12,12,36;
.4,12,12,36,12,36,36,108;
.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324;
.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324,12,36,36,108,36,108,...
The rows converge to A161411. (End)

D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.
S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
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.]
David Applegate, The movie version
Omar E. Pol, Illustration of initial terms (Fig. 1: one-step rook), (Fig. 2: one-step bishop), (Fig. 3: overlapping squares), (Fig. 4: overlapping X-toothpicks), 2009
Omar E. Pol, Illustration of initial terms of A139251, A160121, A147582 (Overlapping figures), 2009
D. Singmaster, On the cellular automaton of Ulam and Warburton, 2003 [Cached copy, included with permission]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021

Cf. A147562, A147610 (the sequence divided by 4), A048881, A000120.
Cf. A000079, A161411, A151779, A139250.
Cf. A048883, A139251, A160121, A162349. [Omar E. Pol, Nov 02 2009]
Cf. A323651.

A000120 := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120; A147582 := n-> if n <= 1 then n else 4*3^(wt(n-1)-1); fi; [seq(A147582(n),n=0..1000)]; # N. J. A. Sloane, Apr 07 2010

s = Plus @@ Flatten@ # & /@ CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 200]; f[n_] = If[n == 0, 1, s[[n + 1]] - s[[n]]]; Array[f, 120, 0] (* Michael De Vlieger, Apr 09 2015, after Nadia Heninger and N. J. A. Sloane at A147562 *)

a(1) = 1; for n > 1, a(n) = 4*3^(wt(n-1)-1) where wt() = A000120(). - R. J. Mathar, Apr 30 2009
This formula is (essentially) given by Singmaster. - N. J. A. Sloane, Aug 06 2009
G.f.: x + 4*x*(Product_{k >= 0} (1 + 3*x^(2^k)) - 1)/3. - N. J. A. Sloane, Jun 10 2009

Extended by R. J. Mathar, Apr 30 2009

A161415 First differences of A160414.

Original entry on oeis.org

1, 8, 12, 28, 12, 36, 36, 92, 12, 36, 36, 108, 36, 108, 108, 292, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 908, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 972, 36, 108, 108, 324, 108, 324, 324, 972, 108, 324, 324

Offset: 1

Omar E. Pol, May 20 2009, Jun 13 2009

nonn

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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Omar E. Pol, Illustration of initial terms [From _Omar E. Pol_, Nov 11 2009]
D. Applegate, Omar E. Pol, N. J. A. Sloane, The toothpick sequence and other sequences from cellular automata, arXiv:1004.3036 [math.CO] [From _R. J. Mathar_, Oct 16 2010]

Cf. A139250, A139251, A160411, A160413, A160414, A160417.
Cf. A160727.
Cf. A048883, A161411, A162349. [From Omar E. Pol, Nov 11 2009]

Contribution from R. J. Mathar, Oct 16 2010: (Start)
isA000079 := proc(n) if type(n,'even') then nops(numtheory[factorset](n)) = 1 ; else false ; fi ; end proc:
A048883 := proc(n) 3^wt(n) ; end proc:
A161415 := proc(n) if n = 1 then 1; elif isA000079(n) then 4*A048883(n-1)-2*n ; else 4*A048883(n-1) ; end if; end proc: seq(A161415(n),n=1..90) ; (End)

a[1] = 1; a[n_] := 4*3^DigitCount[n-1, 2, 1] - If[IntegerQ[Log[2, n]], 2n, 0];
Array[a, 60] (* Jean-François Alcover, Nov 17 2017, after N. J. A. Sloane *)

For n > 1, a(n) = 4*A048883(n-1), except a(n) = 4*A048883(n-1) - 2n if n is a power of 2. - N. J. A. Sloane, Jul 13 2009

More terms from R. J. Mathar, Oct 16 2010

A160412 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 3, 12, 21, 48, 57, 84, 111, 192, 201, 228, 255, 336, 363, 444, 525, 768, 777, 804, 831, 912, 939, 1020, 1101, 1344, 1371, 1452, 1533, 1776, 1857, 2100, 2343, 3072, 3081, 3108, 3135, 3216, 3243, 3324, 3405, 3648, 3675, 3756, 3837, 4080, 4161, 4404, 4647

Offset: 0

Omar E. Pol, May 20 2009, Jun 01 2009

nonn

From Omar E. Pol, Nov 10 2009: (Start)
On the infinite square grid, consider the outside corner of an infinite square.
We start at round 0 with all cells in the OFF state.
The rule: A cell in turned ON iff exactly one of its four vertices is a corner vertex of the set of ON cells. So in each generation every exposed vertex turns on three new cells.
At round 1, we turn ON three cells around the corner of the infinite square, forming a concave-convex hexagon with three exposed vertices.
At round 2, we turn ON nine cells around the hexagon.
At round 3, we turn ON nine other cells. Three cells around of every corner of the hexagon.
And so on.
Shows a fractal-like behavior similar to the toothpick sequence A153006.
For the first differences see the entry A162349.
For more information see A160410, which is the main entry for this sequence.
(End)

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
...77..77..77..77
...766667..766667
....6556....6556.
....654444444456.
...76643344334667
...77.43222234.77
......44211244...
00000000001244...
00000000002234.77
00000000004334667
0000000000444456.
0000000000..6556.
0000000000.766667
0000000000.77..77
0000000000.......
0000000000.......
0000000000.......

Michael De Vlieger, Table of n, a(n) for n = 0..1000
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.]
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 31.
Omar E. Pol, Illustration of initial terms [From _Omar E. Pol_, Nov 10 2009]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata - _Omar E. Pol_, Nov 10 2009

Cf. A139250, A139251, A153006, A152980, A160410, A160414.

Cf. A130665, A162349. - Omar E. Pol, Nov 10 2009

a[n_] := 3*Sum[3^DigitCount[k, 2, 1], {k, 0, n - 1}]; Array[a, 48, 0] (* Michael De Vlieger, Nov 01 2022 *)

From Omar E. Pol, Nov 10 2009: (Start)
a(n) = A160410(n)*3/4.
a(0) = 0, a(n) = A130665(n-1)*3, for n>0.
(End)

More terms from Omar E. Pol, Nov 10 2009
Edited by Omar E. Pol, Nov 11 2009
More terms from Nathaniel Johnston, Nov 06 2010
More terms from Colin Barker, Apr 19 2015

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A147582 First differences of A147562.

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A161415 First differences of A160414.

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A160412 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

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