A160412 -id:A160412 - cp's OEIS frontend

A162349 First differences of A160412.

3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 729, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 729, 27, 81, 81, 243, 81, 243, 243, 729, 81, 243, 243, 729, 243, 729

Offset: 1

Omar E. Pol, Jul 14 2009

nonn

Note that if A048883 is written as a triangle then rows converge to this sequence. - Omar E. Pol, Nov 15 2009

Omar E. Pol, Illustration of initial terms (Neighbors of the vertices).

Cf. A063787, A152980, A160410, A160411, A160412, A160414, A160415, A160417, A160797.

Cf. A048883, A161415. - Omar E. Pol, Nov 15 2009

a[n_] := 3^(1 + DigitCount[n - 1, 2, 1]); Array[a, 100] (* Amiram Eldar, Feb 02 2024 *)

a(n) = 3^A063787(n) = 3 * A048883(n-1). - Amiram Eldar, Feb 02 2024

More terms from Omar E. Pol, Nov 15 2009
More terms from Colin Barker, Apr 19 2015
More terms from Amiram Eldar, Feb 02 2024

A130665 a(n) = Sum_{k=0..n} 3^wt(k), where wt() = A000120().

Original entry on oeis.org

1, 4, 7, 16, 19, 28, 37, 64, 67, 76, 85, 112, 121, 148, 175, 256, 259, 268, 277, 304, 313, 340, 367, 448, 457, 484, 511, 592, 619, 700, 781, 1024, 1027, 1036, 1045, 1072, 1081, 1108, 1135, 1216, 1225, 1252, 1279, 1360, 1387, 1468, 1549, 1792, 1801, 1828, 1855

Offset: 0

N. J. A. Sloane, based on a message from Don Knuth, Jun 23 2007

Partial sums of A048883. - David Applegate, Jun 11 2009
From Gary W. Adamson, Aug 26 2016: (Start)
The formula of Mar 26 2010 is equivalent to the left-shifted vector of matrix powers (lim_{k->infinity} M^k), of the production matrix M:
  1, 0, 0, 0, 0, 0, ...
  4, 0, 0, 0, 0, 0, ...
  3, 1, 0, 0, 0, 0, ...
  0, 4, 0, 0, 0, 0, ...
  0, 3, 1, 0, 0, 0, ...
  0, 0, 4, 0, 0, 0, ...
  0, 0, 3, 1, 0, 0, ...
  ...
The sequence divided by its aerated variant is (1, 4, 3, 0, 0, 0, ...). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, The movie version
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, pp. 27, 31-32.
Tanya Khovanova and Joshua Xiong, Nim Fractals, arXiv:1405.594291 [math.CO], 2014 and J. Int. Seq. 17 (2014) # 14.7.8.
D. E. Knuth, Problem 11320 Amer. Math. Monthly, Vol. 114 No. 9 (Nov 2007) p. 835.
Omar E. Pol, Illustration of initial terms: Fig. 1. Neighbors of the vertices, Fig. 2. Overlapping squares, Fig. 3. One-step bishop, (Nov 08 2009)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Mike Warburton, Ulam-Warburton Automaton - Counting Cells with Quadratics, arXiv:1901.10565 [math.CO], 2019.

Cf. A000120, A006046 (Sum 2^wt(n)), A116520, A130667, A147562, A151920, A151922, A160410, A160412, A360189.

a130665 = sum . map (3 ^) . (`take` a000120_list) . (+ 1)
-- Reinhard Zumkeller, Apr 18 2012

u:=3; a[1]:=1; M:=30; for n from 1 to M do a[2*n] := (u+1)*a[n]; a[2*n+1] := u*a[n] + a[n+1]; od; t1:=[seq( a[n], n=1..2*M )]; # Gives sequence with a different offset

f[n_] := Sum[3^Count[ IntegerDigits[k, 2], 1], {k, 0, n}]; Array[f, 51, 0] (* Robert G. Wilson v, Jun 28 2010 *)

def a(n):  # formula version, n=10^10000 takes ~1 second
    if n == 0:
        return 1
    msb = 1 << (n.bit_length() - 1)
    return msb**2 + 3 * a(n-msb) # Stefan Pochmann, Mar 15 2023

def a(n):  # optimized, n=10^50000 takes ~1 second
    n += 1
    total = 0
    power3 = 1
    while n:
        log = n.bit_length() - 1
        total += power3 << (2*log)
        n -= 1 << log
        power3 *= 3
    return total # Stefan Pochmann, Mar 15 2023

With a different offset: a(1) = 1; a(n) = max { 3*a(k)+a(n-k) | 1 <= k <= n/2 }, for n>1.
a(2n+1) = 4*a(n) and a(2n) = 3*a(n-1) + a(n).
a(n) = (A147562(n+1) - 1)*3/4 + 1. - Omar E. Pol, Nov 08 2009
a(n) = A160410(n+1)/4. - Omar E. Pol, Nov 12 2009
Let r(x) = (1 + 4x + 3x^2), then (1 + 4x + 7x^2 + 16x^3 + ...) =
r(x)* r(x^2) * r(x^4) * r(x^8) * ... - Gary W. Adamson, Mar 26 2010
For asymptotics see the discussion in the comments in A006046. - N. J. A. Sloane, Mar 11 2021
a(n) = Sum_{k=0..floor(log_2(n+1))} 3^k * A360189(n,k). - Alois P. Heinz, Mar 06 2023
a(n) = msb^2 + 3*a(n-msb), where msb = A053644(n). - Stefan Pochmann, Mar 15 2023

Simpler definition (and new offset) from David Applegate, Jun 11 2009

Lower limit of sum in definition changed from 1 to 0 by Robert G. Wilson v, Jun 28 2010

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

A160720 Number of "ON" cells at n-th stage in 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 5, 9, 21, 25, 37, 49, 77, 81, 93, 105, 133, 145, 173, 201, 261, 265, 277, 289, 317, 329, 357, 385, 445, 457, 485, 513, 573, 601, 661, 721, 845, 849, 861, 873, 901, 913, 941, 969, 1029, 1041, 1069, 1097, 1157, 1185, 1245, 1305, 1429, 1441, 1469, 1497

Offset: 0

Omar E. Pol, May 25 2009

nonn

We work on the vertices of the square grid Z^2, and define the neighbors of a cell to be the four closest cells along the diagonals.
We start at stage 0 with all cells in OFF state.
At stage 1, we turn ON a single cell at the origin.
Once a cell is ON it stays ON.
At each subsequent stage, a cell in turned ON if exactly one of its neighboring cells that are no further from the origin is ON.
The "no further from the origin" condition matters for the first time at stage 8, when only A160721(8) = 28 cells are turned ON, and a(8) = 77. In contrast, A147562(8) = 85, A147582(8) = 36.
This CA also arises as the cross-section in the (X,Y)-plane of the CA in A151776.
In other words, a cell is turned ON if exactly one of its vertices touches an exposed vertex of a ON cell of the previous generation. A special rule for this sequence is that every ON cell has only one vertex that should be considered not exposed: its nearest vertex to the center of the structure.
Analog to the "outward" version (A266532) of the Y-toothpick cellular automaton of A160120 on the triangular grid, but here we have ON cells on the square grid. See also the formula section. - Omar E. Pol, Jan 19 2016
This cellular automaton can be interpreted as the outward version of the Ulam-Warburton two-dimensional cellular automaton (see A147562). - Omar E. Pol, Jun 22 2017

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
9...............9
.8.8.8.8.8.8.8.8.
..7...7...7...7..
.8.6.6.....6.6.8.
....5.......5....
.8.6.4.4.4.4.6.8.
..7...3...3...7..
.8...4.2.2.4...8.
........1........
.8...4.2.2.4...8.
..7...3...3...7..
.8.6.4.4.4.4.6.8.
....5.......5....
.8.6.6.....6.6.8.
..7...7...7...7..
.8.8.8.8.8.8.8.8.
9...............9

JungHwan Min, Table of n, a(n) for n = 0..5000
David Applegate, The movie version
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. 30.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A147562, A159912, A160117, A160118, A160120, A160721, A160740, A147562, A160410, A160414, A160412, A160416, A160796, A266532, A267700.

cellOn := [[0,0]] : bbox := [0,0,0,0]: # llx, lly, urx, ury isOn := proc(x,y,L) local i ; for i in L do if op(1,i) = x and op(2,i) = y then RETURN(true) ; fi; od: RETURN(false) ; end: bb := proc(L) local mamin,i; mamin := [0,0,0,0] ; for i in L do mamin := subsop(1=min(op(1,mamin),op(1,i)),mamin) ; mamin := subsop(2=min(op(2,mamin),op(2,i)),mamin) ; mamin := subsop(3=max(op(1,mamin),op(1,i)),mamin) ; mamin := subsop(4=max(op(2,mamin),op(2,i)),mamin) ; od: mamin ; end: for gen from 2 to 80 do nGen := [] ; print(nops(cellOn)) ; for x from op(1,bbox)-1 to op(3,bbox)+1 do for y from op(2,bbox)-1 to op(4,bbox)+1 do # not yet in list? if not isOn(x,y,cellOn) then
# loop over 4 neighbors of (x,y) non := 0 ; for dx from -1 to 1 by 2 do for dy from -1 to 1 by 2 do # test of a neighbor nearer to origin if x^2+y^2 >= (x+dx)^2+(y+dy)^2 then if isOn(x+dx,y+dy,cellOn) then non := non+1 ; fi; fi; od: od: # exactly one neighbor on: add to nGen if non = 1 then nGen := [op(nGen), [x,y]] ; fi; fi; od: od: # merge old and new generation cellOn := [op(cellOn),op(nGen)] ; bbox := bb(cellOn) ; od: # R. J. Mathar, Jul 14 2009

A160720[0]=0; A160720[n_]:=Total[With[{m = n - 1}, BitOr @@ (Function[pos, CellularAutomaton[{FromDigits[Boole[#[[2, 2]] == 1 || Count[Flatten[#], 1] == 1 && Count[Extract[#, pos], 1] == 1] & /@ Tuples[{1, 0}, {3, 3}], 2], 2, {1, 1}}, {{{1}}, 0}, {{{m}}, {-m, m}, {-m, m}}]] /@ Partition[{{-1, -1}, {-1, 1}, {1, 1}, {1, -1}}, 2, 1, 1])], 2] (* JungHwan Min, Jan 23 2016 *)
A160720[0]=0; A160720[n_]:=Total[With[{m = n - 1}, BitOr @@ (CellularAutomaton[{#, 2, {1, 1}}, {{{1}}, 0}, {{{m}}, {-m, m}, {-m, m}}] & /@ {13407603346151304507647333602124270744930157291580986197148043437687863763597662002711256755796972443613438635551055889478487182262900810351549134401372178, 13407603346151304507647333602124270744930157291580986197148043437687863763597777794800494071992396014598447323458909159463152822826940267935557047531012112, 13407603346151304507647333602124270744930157291580986197148043437687863763597777794800494071992396014598447323458909159463152822826940286382301121240563712, 13407603346151304507647333602124270744930157291580986197148043437687863763597662002711256755796972443613438635551055889478487182262900828798293208110923778})], 2] (* JungHwan Min, Jan 23 2016 *)
A160720[0]=0; A160720[n_]:=Total[With[{m = n - 1}, BitOr @@ (CellularAutomaton[{46, {2, ReplacePart[ArrayPad[{{1}}, 1], # -> 2]}, {1, 1}}, {{{1}}, 0}, {{{m}}, All, All}] & /@ Partition[{{-1, -1}, {-1, 1}, {1, 1}, {1, -1}}, 2, 1, 1])], 2] (* JungHwan Min, Jan 24 2016 *)

Conjecture: a(n) = 1 + 4*(A266532(n) - 1)/3, n >= 1. - Omar E. Pol, Jan 19 2016. This formula is correct! - N. J. A. Sloane, Jan 23 2016

a(n) = 1 + 4*A267700(n-1) = 1 + 2*(A159912(n) - n), n >= 1. - Omar E. Pol, Jan 24 2016

Edited by N. J. A. Sloane, Jun 26 2009

More terms from David Applegate, Jul 03 2009

A160796 Total number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton which is the "corner" structure corresponding to A160118.

Original entry on oeis.org

0, 1, 8, 11, 32, 35, 56, 65, 128, 131, 152, 161, 224, 233, 296, 323, 512, 515, 536, 545, 608, 617, 680, 707, 896, 905, 968, 995, 1184, 1211, 1400, 1481, 2048, 2051, 2072, 2081, 2144, 2153, 2216, 2243, 2432, 2441, 2504, 2531, 2720, 2747, 2936, 3017, 3584, 3593, 3656

Offset: 0

Omar E. Pol, Jun 13 2009, Jun 14 2009

nonn

This bears the same relationship to A160118 as A153006 does to A139250.

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
..9...............9
...888.888.888.888.
...878.878.878.878.
...8866688.8866688.
.....656.....656...
...8866444.4446688.
...878.434.434.878.
...888.4422244.888.
.........212.......
00000000002244.888.
0000000000.434.878.
0000000000.4446688.
0000000000...656...
0000000000.8866688.
0000000000.878.878.
0000000000.888.888.
0000000000........9
0000000000.........
0000000000.........

Cf. A139250, A153006, A160797, A160798, A160412, A160118, A160416.

With[{d = 2}, wt[n_] := DigitCount[n, 2, 1]; a[n_] := (5 + 3 * If[OddQ[n], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 1)/2}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 3)/2}], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}]]) / 4; a[0] = 0; a[1] = 1; Array[a, 50, 0]] (* Amiram Eldar, Aug 01 2023 *)

a(n) = 2 + (3/4)*(A160118(n) - 1) if n >= 2.

Entry revised by Omar E. Pol and N. J. A. Sloane, Feb 16 2010
More terms from Nathaniel Johnston, Nov 13 2010
Corrected by Sean A. Irvine, Mar 23 2011, in response to correction to A160118
More terms from Amiram Eldar, Aug 01 2023

A160416 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, 1, 8, 11, 32, 39, 80, 89, 146, 159

Offset: 0

Omar E. Pol, May 20 2009, Jun 14 2009

nonn

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
..9...9...9...9...9
...888.888.888.888.
...878.878.878.878.
...886668666866688.
..9..656.656.656..9
...886644464446688.
...878.434.434.878.
...886644222446688.
..9..656.212.656..9
000000000022446688.
0000000000.434.878.
000000000064446688.
000000000056.656..9
000000000066866688.
0000000000.878.878.
0000000000.888.888.
00000000009...9...9
0000000000.........
0000000000.........

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

Cf. A139250, A139251, A152980, A153006, A160117, A160118, A160410, A160412, A160414.

Cf. A160796, A161417. [From Omar E. Pol, Jun 14 2009]

cp's OEIS Frontend

A162349 First differences of A160412.

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A130665 a(n) = Sum_{k=0..n} 3^wt(k), where wt() = A000120().

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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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A160720 Number of "ON" cells at n-th stage in 2-dimensional cellular automaton (see Comments for precise definition).

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A160796 Total number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton which is the "corner" structure corresponding to A160118.

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A160416 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

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