A048883 -id:A048883 - cp's OEIS frontend

A161415 First differences of A160414.

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

A151710 First differences of A160715.

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 9, 21, 3, 9, 9, 21, 15, 21, 27, 51, 3, 9, 9, 21, 15, 21, 27, 51, 21, 21, 27, 51

Offset: 1

Omar E. Pol, Jun 02 2009

Number of Y-toothpick added at n-th stage to the structure of A160715.

Similar to A160121. [From Omar E. Pol, May 29 2010]

			Contribution from _Omar E. Pol_, Dec 18 2012 (Start):
Written as an irregular triangle begins:
1;
3;
3, 9;
3, 9, 9, 21;
3, 9, 9, 21, 15, 21, 27, 51;
3, 9, 9, 21, 15, 21, 27, 51, 21, 21, 27, 51,...
(End)

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. A000225, A006516, A139250, A160120.

Cf. A048883, A139251, A151837, A160121.

More terms from Omar E. Pol, May 29 2010

A116640 a(n) = A116623(A059893(n)).

Original entry on oeis.org

1, 5, 7, 19, 11, 23, 29, 65, 19, 31, 37, 73, 49, 85, 103, 211, 35, 47, 53, 89, 65, 101, 119, 227, 89, 125, 143, 251, 179, 287, 341, 665, 67, 79, 85, 121, 97, 133, 151, 259, 121, 157, 175, 283, 211, 319, 373, 697, 169, 205, 223, 331, 259, 367, 421, 745, 331

Offset: 0

Antti Karttunen, Feb 20 2006. Proposed by Pierre Lamothe (plamothe(AT)aei.ca), May 21 2004

Viewed as a binary tree, this is (1); 5; 7,19; 11,23,29,65; ... Cf. A116623.

If we treat (2n+1) as a binary number with the nonzero bits numbered (highest bit first) from 0..k and the regular binary place value of each nonzero bit numbered from b(0) to b(k) then a(n) = 3^0 * b(0) + 3^1 * b(1) + .. + 3^k. For instance, if n=6 then 2n+1 = 13, which is equal to 8+4+1 or 1101 base(2); and a(n)=29 which is 8*1 + 4*3 + 1*9. - Joe Slater, Jan 23 2016

Cf. A000079, A000225, A001047, A062709.

a:= proc(n) option remember; piecewise(
    n mod 4 = 0, 3*procname(n/2) - 2*procname(n/4),
  n mod 4 = 1, 6*procname((n-1)/4) - procname((n-1)/2),
  n mod 4 = 2, procname(n/2) + 2*procname((n-2)/4),
  5*procname((n-1)/2) - 6*procname((n-3)/4))
end proc:
a(0):= 1:
map(a, [$0..100]); # Robert Israel, Jan 19 2016

a[n_] := a[n] = Switch[Mod[n, 4], 0, 3a[Floor[n/2]] - 2a[Floor[n/4]], 1, 6a[Floor[n/4]] - a[Floor[n/2]], 2, a[Floor[n/2]] + 2a[Floor[n/4]], 3, 5a[Floor[n/2]] - 6a[Floor[n/4]]]; a[0]=1; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 28 2016 *)

a(n) = if(n==0, return(1)); 2*a(n\2) - (-1)^n * 3^hammingweight(n) \\ Charles R Greathouse IV, Jan 21 2016

a(n) = my(p=2*n+1,v=vecextract(vector(#binary(p),j,2^(j-1)),p));sum(i=0,#v-1,3^i*v[#v-i]) \\ Joe Slater, May 09 2017

a(A000225(n)) = A001047(n+1).
For n>= 1 a(A000079(n)) = A062709(n+1).
From Joe Slater, Jan 19 2016: (Start)
a(0) = 1,
a(n) = 3*a(floor(n/2)) - 2*a(floor(n/4)) for n=0 (mod 4) and n>0,
a(n) = 6*a(floor(n/4)) - a(floor(n/2)) for n=1 (mod 4),
a(n) = a(floor(n/2)) + 2*a(floor(n/4)) for n=2 (mod 4),
a(n) = 5*a(floor(n/2)) - 6*a(floor(n/4)) for n=3 (mod 4)
(End)
a(0) = 1, a(n) = 2*a(floor(n/2)) - A033999(n) * A048883(n) for n>0. -
Joe Slater, Jan 22 2016

A122108 a(n)=number of 1's ("live" cells) at stage n of a 2-dimensional cellular automaton evolving by the rule: 1 if N+S+E+W is 1 or 2, else 0, starting from a single live cell.

Original entry on oeis.org

1, 4, 8, 12, 20, 24, 44, 28, 52, 56, 104, 56, 104, 112, 212, 60, 116, 120, 232, 120, 232, 240, 464, 120, 232, 240, 464, 240, 464, 480, 932, 124, 244, 248, 488, 248, 488, 496, 976, 248, 488, 496, 976, 496, 976, 992, 1952, 248, 488, 496, 976, 496, 976, 992, 1952

Offset: 1

John W. Layman, Oct 18 2006

nonn

Index entries for sequences related to cellular automata

Cf. A048883.

A160727 a(n) = A161415(n+1)/4.

Original entry on oeis.org

2, 3, 7, 3, 9, 9, 23, 3, 9, 9, 27, 9, 27, 27, 73, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 227, 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, 697, 3, 9, 9, 27, 9

Offset: 1

Omar E. Pol, Jun 13 2009

			From _Omar E. Pol_, Jan 01 2014: (Start)
Written as an irregular triangle in which row lengths is A000079 the sequence begins:
2;
3,7;
3,9,9,23;
3,9,9,27,9,27,27,73;
3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,227;
3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27, 81,81,243,27,81,81,243,81,243,243,697;
(End)

Paolo Xausa, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A152978, A153000, A160414, A161415.

A160727[n_]:=3^DigitCount[n,2,1]-If[IntegerQ[Log2[n+1]],(n+1)/2,0];Array[A160727,100] (* Paolo Xausa, Sep 01 2023 *)

a(n) = A048883(n), except a(n) = A048883(n) - (n+1)/2 if n is a power of 2 minus 1. - Omar E. Pol, Jan 06 2014

a(11)-a(58) from M. F. Hasler, Dec 03 2012

a(59)-a(68) from Omar E. Pol, Jan 06 2014

A173453 a(n) = A160121(n) - A151710(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 78

Offset: 1

Omar E. Pol, May 29 2010

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. A048883, A139250, A139251, A160121, A151710, A173066, A173067, A173068, A173451, A173452.

A256140 Square array read by antidiagonals upwards: T(n,k) = n^A000120(k), n>=0, k>=0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 3, 4, 1, 0, 1, 5, 4, 9, 2, 1, 0, 1, 6, 5, 16, 3, 4, 1, 0, 1, 7, 6, 25, 4, 9, 4, 1, 0, 1, 8, 7, 36, 5, 16, 9, 8, 1, 0, 1, 9, 8, 49, 6, 25, 16, 27, 2, 1, 0, 1, 10, 9, 64, 7, 36, 25, 64, 3, 4, 1, 0, 1, 11, 10, 81, 8, 49, 36, 125, 4, 9, 4, 1, 0, 1, 12, 11, 100, 9, 64, 49, 216, 5, 16, 9, 8, 1, 0

Offset: 0

Omar E. Pol, Mar 16 2015

The partial sums of row n give the n-th row of the square array A256141.

First differs from A244003 at a(25).

			The corner of the square array with the first 16 terms of the first 12 rows looks like this:
---------------------------------------------------------------------------
A000007: 1, 0, 0,  0,  0,  0,  0,   0,  0,  0,  0,   0,  0,   0,   0,    0
A000012: 1, 1, 1,  1,  1,  1,  1,   1,  1,  1,  1,   1,  1,   1,   1,    1
A001316: 1, 2, 2,  4,  2,  4,  4,   8,  2,  4,  4,   8,  4,   8,   8,   16
A048883: 1, 3, 3,  9,  3,  9,  9,  27,  3,  9,  9,  27,  9,  27,  27,   81
A102376: 1, 4, 4, 16,  4, 16, 16,  64,  4, 16, 16,  64, 16,  64,  64,  256
A256135: 1, 5, 5, 25,  5, 25, 25, 125,  5, 25, 25, 125, 25, 125, 125,  625
A256136: 1, 6, 6, 36,  6, 36, 36, 216,  6, 36, 36, 216, 36, 216, 216, 1296
.......: 1, 7, 7, 49,  7, 49, 49, 343,  7, 49, 49, 343, 49, 343, 343, 2401
.......: 1, 8, 8, 64,  8, 64, 64, 512,  8, 64, 64, 512, 64, 512, 512, 4096
.......: 1, 9, 9, 81,  9, 81, 81, 729,  9, 81, 81, 729, 81, 729, 729, 6561
.......: 1,10,10,100, 10,100,100,1000, 10,100,100,1000,100,1000,1000,10000
.......: 1,11,11,121, 11,121,121,1331, 11,121,121,1331,121,1331,1331,14641

Index entries for sequences related to cellular automata

Cf. A000120, A244003, A255740, A255741, A256141.
First seven rows are A000007, A000012, A001316, A048883, A102376, A256135, A256136.
First 16 columns are A000012, A001477, A001477, A000290, A001477, A000290, A000290, A000578, A001477, A000290, A000290, A000578, A000290, A000578, A000578, A000583.
Main diagonal gives A302451.

A164982 Number of ON cells after n generations of the 2D cellular automaton described in the comments.

Original entry on oeis.org

1, 3, 4, 12, 7, 21, 16, 40, 22, 42, 34, 67, 52, 85, 70, 125, 94, 126, 102, 150, 118, 172, 177, 234, 209, 240, 238, 319, 285, 363, 378, 458, 383, 444, 404, 493, 474, 520, 529, 628, 583, 602, 622, 727, 664, 816, 835, 948, 873, 926, 952, 1065, 1010, 1090, 1187

Offset: 1

John W. Layman, Sep 03 2009

nonn

The cells are the squares of the standard square grid. All cells are initially OFF and one cell is turned ON at generation 1. At subsequent generations a cell is ON if and only if (1) exactly one of neighbors NW, NE, and S was ON, or (2) all three of cells N, SW, and SE were ON in the previous generation. (The 9-cell Moore neighborhood is labeled {{NW,N,NE},{W,C,E},{SW,S,SE}}).

John W. Layman, Graphs of the automaton for generations 1-20 [From _John W. Layman_, Sep 14 2009]
Index entries for sequences related to cellular automata

Cf. A048883, A072272, A122108, A160410, A247002.

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=61986;
Show[GraphicsArray[Map[RasterGraphics, CellularAutomaton[{rule, {2, {{1, 4, 1}, {0, 0, 0}, {4, 1, 4}}}, {1, 1}}, {{{1}}, 0}, 4, -5]]]];
ca = CellularAutomaton[{rule, {2, {{1, 4, 1}, {0, 0, 0}, {4, 1, 4}}}, {1, 1}}, {{{1}}, 0}, 99, -100];
Table[Total[ca[[i]], 2], {i, 1, 100}]

A119733 Offsets of the terms of the nodes of the reverse Collatz function.

Original entry on oeis.org

0, 1, 2, 5, 4, 7, 10, 19, 8, 11, 14, 23, 20, 29, 38, 65, 16, 19, 22, 31, 28, 37, 46, 73, 40, 49, 58, 85, 76, 103, 130, 211, 32, 35, 38, 47, 44, 53, 62, 89, 56, 65, 74, 101, 92, 119, 146, 227, 80, 89, 98, 125, 116, 143, 170, 251, 152, 179, 206, 287, 260, 341, 422, 665, 64, 67

Offset: 0

William Entriken, Jun 14 2006

nonn

Create a binary tree starting with x. To follow 0 from the root, apply f(x)=2x. To follow 1, apply g(x)=(2x-1)/3. For example, starting with x, the string 010 {also known as f(g(f(x)))}, you would get (8x-2)/3. These expressions represent the reverse Collatz function and will provide numbers whose Collatz path may include x. These expressions will all be of the form (2^a*x-b)/3^c. This sequence concerns b. What makes b interesting is that if you draw the tree, each level of the tree will have the same sequence of values for b. The root of the tree x, can be written as (2^0*x-0)/3^0, which has the first value for b. Each subsequent level contains twice as many values of b.
This sequence is 0 followed by a permutation of A213539, and therefore consists of 0 plus the elements of A116640 multiplied by 2^k, where k >= 0. E.g., 1, 5, 7, 19 becomes 0, 2^0*1, 2^1*1, 2^0*5, 2^2*1, 2^0*7, 2^1*5, 2^0*19, ... - Joe Slater, Dec 19 2016
When this sequence is arranged as an irregular triangle the sum of each row a(2^k)...a(2^(k+1)-1) equals A081039(2^k). The cumulative sum from a(0) to a(2^k-1) equals A002697(k). - Joe Slater, Apr 12 2018

			a(1) = 1 = 2 * 0 + 3^0 since 0 written in binary contains no 1's.

Alois P. Heinz, Table of n, a(n) for n = 0..16383
Víctor Martín Chabrera, An algebraic fractal approach to Collatz Conjecture, Bachelor tesis, Universitat Politècnica de Catalunya (Barcelona, 2019), see lemma 6.1 with a(n) = r(A005836(n)).

Cf. A002697, A081039, A116623, A116640, A116641, A213539, A226383.

a:= proc(n) `if`(n=0, 0, `if`(irem(n, 2, 'r')=0, 0,
      3^add(i, i=convert(r, base, 2)))+2*a(r))
    end:
seq(a(n), n=0..127);  # Alois P. Heinz, Aug 13 2017

a[0] := 0; a[n_?OddQ] := 2a[(n - 1)/2] + 3^Plus@@IntegerDigits[(n - 1)/2, 2]; a[n_?EvenQ] := 2a[n/2]; Table[a[n], {n, 0, 65}] (* Alonso del Arte, Apr 21 2011 *)

a(n) = my(ret=0); if(n, for(i=0,logint(n,2), if(bittest(n,i), ret=3*ret+1<Kevin Ryde, Oct 22 2021

# call with n to get 2^n values
$depth=shift; sub funct { my ($i, $b, $c) = @_; if ($i < $depth) { funct($i+1, $b*2, $c); funct($i+1, 2*$b+$c, $c*3); } else { print "$b, "; } } funct(0, 0, 1); print " ";

from sympy.core.cache import cacheit
@cacheit
def a(n): return 0 if n==0 else 2*a((n - 1)//2) + 3**bin((n - 1)//2).count('1') if n%2 else 2*a(n//2)
print([a(n) for n in range(131)]) # Indranil Ghosh, Aug 13 2017

a(0) = 0, a(2*n + 1) = 2*a(n) + 3^wt(n) = 2*a(n) + A048883(n), a(2*n) = 2*a(n), where wt(n) = A000120(n) = the number 1's in the binary representation of n.
a(k) = [z^k] 1 + (1/(1-z)) * Sum_{s=0..n-1} 2^s*z^(2^s)*(1 - z^(2^s)) * Product_{r=s+1..n-1} (1 + 3*z^(2^r)), for 0 < k <= 2^n-1. - Wolfgang Hintze, Jul 28 2017
a(n) = Sum_{i=0..k} 2^e[i] * 3^i where binary expansion n = 2^e[0] + 2^e[1] + ... + 2^e[k] with descending e[0] > e[1] > ... > e[k] (A272011). [Martín Chabrera lemma 6.1, adapting index i] - Kevin Ryde, Oct 22 2021

A162349 First differences of A160412.

Original entry on oeis.org

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

cp's OEIS Frontend

A161415 First differences of A160414.

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A151710 First differences of A160715.

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A116640 a(n) = A116623(A059893(n)).

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A122108 a(n)=number of 1's ("live" cells) at stage n of a 2-dimensional cellular automaton evolving by the rule: 1 if N+S+E+W is 1 or 2, else 0, starting from a single live cell.

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A160727 a(n) = A161415(n+1)/4.

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A173453 a(n) = A160121(n) - A151710(n).

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A256140 Square array read by antidiagonals upwards: T(n,k) = n^A000120(k), n>=0, k>=0.

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A164982 Number of ON cells after n generations of the 2D cellular automaton described in the comments.

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Mathematica

A119733 Offsets of the terms of the nodes of the reverse Collatz function.

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