A256534 -id:A256534 - cp's OEIS frontend

A261695 First differences of A256534.

0, 4, 12, 12, 36, 12, 36, 60, 84, 12, 36, 60, 84, 108, 132, 156, 180, 12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 732, 756, 12, 36, 60, 84, 108

Offset: 0

Omar E. Pol, Sep 24 2015

Number of cells turned ON at n-th stage of cellular automaton of A256534.

Similar to A256531 which shares infinitely many terms.

			With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
4;
12;
12, 36;
12, 36, 60, 84;
12, 36, 60, 84, 108, 132, 156, 180;
12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372;
...

Cf. A011782, A139251, A147582, A161411, A161415, A241717, A256531, A256534.

It appears that a(n) = 4 * A241717(n-1), n >= 1.

A160410 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, 4, 16, 28, 64, 76, 112, 148, 256, 268, 304, 340, 448, 484, 592, 700, 1024, 1036, 1072, 1108, 1216, 1252, 1360, 1468, 1792, 1828, 1936, 2044, 2368, 2476, 2800, 3124, 4096, 4108, 4144, 4180, 4288, 4324, 4432, 4540, 4864, 4900, 5008, 5116, 5440, 5548, 5872, 6196

Offset: 0

Omar E. Pol, May 20 2009

On the infinite square grid, we consider cells to be the squares, and we start at round 0 with all cells in the OFF state, so a(0) = 0.
At round 1, we turn ON four cells, forming a square.
The rule for n > 1: 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.
Therefore:
At Round 2, we turn ON twelve cells around the square.
At round 3, we turn ON twelve other cells. Three cells around of every corner of the square.
And so on.
For the first differences see the entry A161411.
Shows a fractal behavior similar to the toothpick sequence A139250.
A very similar sequence is A160414, which uses the same rule but with a(1) = 1, not 4.
When n=2^k then the polygon formed by ON cells is a square with side length 2^(k+1).
a(n) is also the area of the figure of A147562 after n generations if A147562 is drawn as overlapping squares. - Omar E. Pol, Nov 08 2009
From Omar E. Pol, Mar 28 2011: (Start)
Also, toothpick sequence starting with four toothpicks centered at (0,0) as a cross.
Rule: Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of three toothpicks of new generation. (Note that these three toothpicks looks like a T-toothpick, see A160172.)
The sequence gives the number of toothpicks after n stages. A161411 gives the number of toothpicks added at the n-th stage.
(End)

			From _Omar E. Pol_, Sep 24 2015: (Start)
With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
    4;
   16;
   28,  64;
   76, 112, 148, 256;
  268, 304, 340, 448, 484, 592, 700, 1024;
  ...
Right border gives the elements of A000302 greater than 1.
This triangle T(n,k) shares with the triangle A256534 the terms of the column k, if k is a power of 2, for example, both triangles share the following terms: 4, 16, 28, 64, 76, 112, 256, 268, 304, 448, 1024, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                         _ _ _ _
.      |  _ _  |                       |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _     _ _     _ _     _ _  |_| |
.      |_ _| |  _|_ _|_  |   |  _|_ _|_  | |_ _|
.          | |_|  _ _  |_|   |_|  _ _  |_| |
.          |   | |  _|_|_ _ _ _|_|_  | |   |
.          |  _| |_|  _ _     _ _  |_| |_  |
.          | | |_ _| |  _|_ _|_  | |_ _| | |
.          | |_ _| | |_|  _ _  |_| | |_ _| |
.          |       |   | |   | |   |       |
.          |  _ _  |  _| |_ _| |_  |  _ _  |
.          | |  _|_| | |_ _ _ _| | |_|_  | |
.          | |_|  _| |_ _|   |_ _| |_  |_| |
.          |   | | |_ _ _ _ _ _ _ _| | |   |
.          |  _| |_ _| |_     _| |_ _| |_  |
.       _ _| | |_ _ _ _| |   | |_ _ _ _| | |_ _
.      |  _| |_ _|   |_ _|   |_ _|   |_ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                       | |_ _| |
.      |_ _ _ _|                       |_ _ _ _|
.
After 10 generations there are 304 ON cells, so a(10) = 304.
(End)

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 (2009)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000079, A000302, A048883, A139250, A147562, A160118, A160412, A160414, A161411, A160717, A160720, A160727, A256530, A256534.

RasterGraphics[state_?MatrixQ,colors_Integer:2,opts___]:=
Graphics[Raster[Reverse[1-state/(colors -1)]],
AspectRatio ->(AspectRatio/.{opts}/.AspectRatio ->Automatic),
Frame ->True, FrameTicks ->None, GridLines ->None];
rule=1340761804646523638425234105559798690663900360577570370705802859623\
705267234688669629039040624964794287326910250673678735142700520276191850\
5902735959769690
Show[GraphicsArray[Map[RasterGraphics,CellularAutomaton[{rule, {2,
{{4,2,1}, {32,16,8}, {256,128,64}}}, {1,1}}, {{{1,1}, {1,1}}, 0}, 9,-10]]]];
ca=CellularAutomaton[{rule,{2,{{4,2,1},{32,16,8},{256,128,64}}},{1,
1}},{{{1,1},{1,1}},0},99,-100];
Table[Total[ca[[i]],2],{i,1,Length[ca]}]
(* John W. Layman, Sep 01 2009; Sep 02 2009 *)
a[n_] := 4*Sum[3^DigitCount[k, 2, 1], {k, 0, n-1}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 17 2017, after N. J. A. Sloane *)

A160410(n)=sum(i=0,n-1,3^norml2(binary(i)))<<2 \\ M. F. Hasler, Dec 04 2012

Equals 4*A130665. This provides an explicit formula for a(n). - N. J. A. Sloane, Jul 13 2009

a(2^k) = (2*(2^k))^2 for k>=0.

Edited by David Applegate and N. J. A. Sloane, Jul 13 2009

A160414 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (same as A160410, but a(1) = 1, not 4).

Original entry on oeis.org

0, 1, 9, 21, 49, 61, 97, 133, 225, 237, 273, 309, 417, 453, 561, 669, 961, 973, 1009, 1045, 1153, 1189, 1297, 1405, 1729, 1765, 1873, 1981, 2305, 2413, 2737, 3061, 3969, 3981, 4017, 4053, 4161, 4197, 4305, 4413, 4737, 4773, 4881, 4989, 5313, 5421, 5745

Offset: 0

Omar E. Pol, May 20 2009

The structure has a fractal behavior similar to the toothpick sequence A139250.
First differences: A161415, where there is an explicit formula for the n-th term.
For the illustration of a(24) = 1729 (the Hardy-Ramanujan number) see the Links section.

			From _Omar E. Pol_, Sep 24 2015: (Start)
With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
9;
21,    49;
61,    97,  133,  225;
237,  273,  309,  417,  453, 561,  669,  961;
...
Right border gives A060867.
This triangle T(n,k) shares with the triangle A256530 the terms of the column k, if k is a power of 2, for example both triangles share the following terms: 1, 9, 21, 49, 61, 97, 225, 237, 273, 417, 961, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                       _ _ _ _
.      |  _ _  |                     |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _     _ _   _ _     _ _  |_| |
.      |_ _| |  _|_ _|_  | |  _|_ _|_  | |_ _|
.          | |_|  _ _  |_| |_|  _ _  |_| |
.          |   | |  _|_|_ _ _|_|_  | |   |
.          |  _| |_|  _ _   _ _  |_| |_  |
.          | | |_ _| |  _|_|_  | |_ _| | |
.          | |_ _| | |_|  _  |_| | |_ _| |
.          |  _ _  |  _| |_| |_  |  _ _  |
.          | |  _|_| | |_ _ _| | |_|_  | |
.          | |_|  _| |_ _| |_ _| |_  |_| |
.          |   | | |_ _ _ _ _ _ _| | |   |
.          |  _| |_ _| |_   _| |_ _| |_  |
.       _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
.      |  _| |_ _|   |_ _| |_ _|   |_ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                     | |_ _| |
.      |_ _ _ _|                     |_ _ _ _|
.
After 10 generations there are 273 ON cells, so a(10) = 273.
(End)

Paolo Xausa, Table of n, a(n) for n = 0..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.]
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of the structure after 24th stage (contains 1729 ON cells)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A001235, A011541, A011782, A000225, A060867, A139250, A147562, A160117, A160118, A160410, A160412, A161415, A160720, A160727, A151725, A256530, A256534.

read("transforms") ; 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:
A160414 := proc(n) add( A161415(k),k=1..n) ; end proc: seq(A160414(n),n=0..90) ; # R. J. Mathar, Oct 16 2010

A160414list[nmax_]:=Accumulate[Table[If[n<2,n,4*3^DigitCount[n-1,2,1]-If[IntegerQ[Log2[n]],2n,0]],{n,0,nmax}]];A160414list[100] (* Paolo Xausa, Sep 01 2023, after R. J. Mathar *)

my(s=-1, t(n)=3^norml2(binary(n-1))-if(n==(1<Altug Alkan, Sep 25 2015

a(n) = 1 + 4*A219954(n), n >= 1. - M. F. Hasler, Dec 02 2012

a(2^k) = (2^(k+1) - 1)^2. - Omar E. Pol, Jan 05 2013

Edited by N. J. A. Sloane, Jun 15 2009 and Jul 13 2009

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

A256530 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).

Original entry on oeis.org

0, 1, 9, 21, 49, 61, 97, 157, 225, 237, 273, 333, 417, 525, 657, 813, 961, 973, 1009, 1069, 1153, 1261, 1393, 1549, 1729, 1933, 2161, 2413, 2689, 2989, 3313, 3661, 3969, 3981, 4017, 4077, 4161, 4269, 4401, 4557, 4737, 4941, 5169, 5421, 5697, 5997, 6321, 6669, 7041, 7437, 7857, 8301, 8769, 9261, 9777, 10317, 10881, 11469

Offset: 0

Omar E. Pol, Apr 21 2015

On the infinite square grid at stage 0 there are no ON cells, so a(0) = 0.
At stage 1, only one cell is turned ON, so a(1) = 1.
If n is a power of 2 so the structure is a square of side length 2n - 1 that contains (2n-1)^2 ON cells.
The structure grows by the four corners as square waves forming layers of ON cells up the next square structure, and so on (see example).
Note that a(24) = 1729 is also the Hardy-Ramanujan number (see A001235).
Has the same rules as A256534 but here a(1) = 1 not 4.
Has a smoother behavior than A160414 with which shares infinitely many terms (see example).
A256531, the first differences, gives the number of cells turned ON at n-th stage.

			With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
9;
21,    49;
61,    97,  157,  225;
237,  273,  333,  417,  525,  657,  813,  961;
...
Right border gives A060867.
This triangle T(n,k) shares with the triangle A160414 the terms of the column k, if k is a power of 2, for example both triangles share the following terms: 1, 9, 21, 49, 61, 97, 225, 237, 273, 417, 961, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                       _ _ _ _
.      |  _ _  |                     |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _ _ _ _ _   _ _ _ _ _ _  |_| |
.      |_ _| |  _ _ _ _  | |  _ _ _ _  | |_ _|
.          | | |  _ _  | | | |  _ _  | | |
.          | | | |  _|_|_|_|_|_|_  | | | |
.          | | | |_|  _ _   _ _  |_| | | |
.          | | |_ _| |  _|_|_  | |_ _| | |
.          | |_ _ _| |_|  _  |_| |_ _ _| |
.          |  _ _ _|  _| |_| |_  |_ _ _  |
.          | |  _ _| | |_ _ _| | |_ _  | |
.          | | |  _| |_ _| |_ _| |_  | | |
.          | | | | |_ _ _ _ _ _ _| | | | |
.          | | | |_ _| | | | | |_ _| | | |
.       _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
.      |  _| |_ _ _ _ _ _| |_ _ _ _ _ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                     | |_ _| |
.      |_ _ _ _|                     |_ _ _ _|
.
After 10 generations there are 273 ON cells, so a(10) = 273.

Paolo Xausa, Table of n, a(n) for n = 0..8191
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000225, A011782, A001235, A060867, A139250, A147562, A160410, A160414, A256531, A256534.

With[{z=7},Join[{0},Flatten[Array[(2^#-1)^2+12Range[0,2^(#-1)-1]^2&,z]]]] (* Generates 2^z terms *) (* Paolo Xausa, Nov 15 2023, after Omar E. Pol *)

For i = 1 to z: for j = 0 to 2^(i-1)-1: n = n+1: a(n) = (2^i-1)^2 + 3*(2*j)^2: next j: next i

A236305 The number of P-positions in the game of Nim with up to 3 piles, allowing for piles of zero, such that the number of objects in each pile does not exceed n.

Original entry on oeis.org

1, 4, 7, 16, 19, 28, 43, 64, 67, 76, 91, 112, 139, 172, 211, 256, 259, 268, 283, 304, 331, 364, 403, 448, 499, 556, 619, 688, 763, 844, 931, 1024, 1027, 1036, 1051, 1072, 1099, 1132, 1171, 1216, 1267, 1324, 1387, 1456, 1531, 1612, 1699

Offset: 0

Tanya Khovanova and Joshua Xiong, Apr 21 2014

nonn

P-positions in the game of Nim are tuples of numbers with a Nim-Sum equal to zero.
(0,1,1) is considered different from (1,0,1) and (1,1,0).
a(2^n-1) = 2^(2*n).
Partial sums of A241717.
This sequence seems to be A256534(n+1)/4. - Thomas Baruchel, May 15 2018

			If the largest number is 1, then there should be an even number of piles of size 1. Thus, a(1)=4.

Michael De Vlieger, Table of n, a(n) for n = 0..1024
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. 37.
T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 7 and J. Int. Seq. 17 (2014) # 14.7.8.

Cf. A241522 (4 piles), A241523 (5 piles).

Cf. A241717 (first differences).

Table[Length[Select[Flatten[Table[{n, k, BitXor[n, k]}, {n, 0, a}, {k, 0, a}], 1], #[[3]] <= a &]], {a, 0, 100}]

If b = floor(log_2(n)) is the number of digits in the binary representation of n and c = n + 1 - 2^b, then a(n) = 2^(2*b) + 3*c^2.

a(n) = 4^floor(log(n)/log(2)) + 3*(n mod 2^floor(log(n)/log(2)))^2 (conjectured). - Thomas Baruchel, May 15 2018

cp's OEIS Frontend

A261695 First differences of A256534.

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A160410 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

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A160414 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (same as A160410, but a(1) = 1, not 4).

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Maple

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A256530 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).

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A236305 The number of P-positions in the game of Nim with up to 3 piles, allowing for piles of zero, such that the number of objects in each pile does not exceed n.

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Examples

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Mathematica

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