A151735 -id:A151735 - cp's OEIS frontend

A151737 First differences of A151735 (starting at A151735(2)).

1, 2, 1, 5, 1, 5, 5, 11, 1, 5, 5, 11, 7, 15, 19, 23, 1, 5, 5, 11, 7, 15, 19, 23, 7, 15, 21, 29, 29, 49, 59, 47, 1, 5, 5, 11, 7, 15, 19, 23, 7, 15, 21, 29, 29, 49, 59, 47, 7, 15, 21, 29, 29, 49, 61, 53, 29, 51, 71, 87, 107, 157, 163, 95, 1, 5, 5, 11, 7, 15, 19, 23, 7, 15, 21, 29, 29, 49, 59, 47

Offset: 0

N. J. A. Sloane, Jun 15 2009, Jun 16 2009

			If written as a triangle,
1,
2,
1,5,
1,5,5,11,
1,5,5,11,7,15,19,23,
1,5,5,11,7,15,19,23,7,15,21,29,29,49,59,47,
1,5,5,11,7,15,19,23,7,15,21,29,29,49,59,...
the rows approach A151728.

Cf. A151735, A151728.

A151725 Number of ON states after n generations of cellular automaton rule described by the rulestring B1/S012345678.

Original entry on oeis.org

0, 1, 9, 13, 33, 37, 57, 77, 121, 125, 145, 165, 209, 237, 297, 373, 465, 469, 489, 509, 553, 581, 641, 717, 809, 837, 897, 981, 1097, 1213, 1409, 1645, 1833, 1837, 1857, 1877, 1921, 1949, 2009, 2085, 2177, 2205, 2265, 2349, 2465, 2581, 2777, 3013

Offset: 0

David Applegate and N. J. A. Sloane, Jun 13 2009

nonn

A cell is turned ON if exactly one of its eight neighbors is ON. An ON cell remains ON forever.
We start with a single ON cell.
Analog of A147562, which is the case when each cell has only four neighbors.
The equivalent Mathematica cellular automaton is obtained with neighborhood weights {{1,1,1},{1,9,1},{1,1,1}}, rule number 261634, and starting configuration {{1}}. [John W. Layman, Sep 11 2009]
Observation: Visual pattern similar to the toothpick structure (see A139250). [Omar E. Pol, Dec 14 2009]

David Applegate, Table of n, a(n) for n = 0..1000
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.]
N. H. Packard and S. Wolfram, Two-Dimensional Cellular Automata, Journal of Statistical Physics, 38 (1985), 901-946. (See page 920, Figure 7e). Alternative copy
Rémy Sigrist, Illustration of the structure at stage 513
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A151726, A147562, A147582, A151723, A151735, A151747, A151728.

See A151731, A151732, A151733, A151734 for the same CA except that two neighbors must be ON for a cell to turn ON.

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]; wt = {{1,1,1}, {1,9,1}, {1,1,1}}; rule= 261634; init={{1}}; Show[GraphicsArray[Map[RasterGraphics, CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, 9, -10]]]];nx = 100; ca = CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, nx - 1, -nx]; a = Table[Total[ca[[i]], 2], {i, 1, nx}] (* John W. Layman, Sep 11 2009 *)
A151725[0] = 0; A151725[n_] := Total[CellularAutomaton[{174766, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, {{{n - 1}}}], 2]; Array[A151725, 48, 0] (* JungHwan Min, Sep 01 2016 *)
A151725L[n_] := Prepend[Total[#, 2] & /@ CellularAutomaton[{174766, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, n - 1], 0]; A151725L[47] (* JungHwan Min, Sep 01 2016 *)

For a recurrence see the Applegate-Pol-Sloane paper.

Definition clarified by SiYang Hu, May 10 2025

A151747 Except for boundary cases (n <= 3, j = 0, 1, 2^i-1), satisfies a(n) = a(2^i+j) = 2 a(j) + a(j+1), where n = 2^i + j, 0 <= j < 2^i .

Original entry on oeis.org

0, 1, 3, 5, 8, 9, 11, 17, 21, 15, 11, 18, 25, 29, 39, 54, 53, 27, 11, 18, 25, 29, 39, 55, 57, 41, 40, 61, 79, 97, 132, 160, 129, 51, 11, 18, 25, 29, 39, 55, 57, 41, 40, 61, 79, 97, 132, 161, 133, 65, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 424, 448, 305, 99, 11, 18

Offset: 0

David Applegate, Jun 16 2009

The boundary cases are covered by the following formulas:
a(n) = 2n-1 if n<=3.
a(n) = 1+(3*i+1)*2^(i-2) if j=0.
a(n) = 3+ 3*2^(i-1) if j= 1.
a(n) = 2*a(j)+a(j+1)-1 if j=2^i-1.

			If written as a triangle:
.0,
.1,
.3, 5,
.8, 9, 11, 17,
.21, 15, 11, 18, 25, 29, 39, 54,
.53, 27, 11, 18, 25, 29, 39, 55, 57, 41, 40, 61, 79, 97, 132, 160,
.129, 51, 11, 18, 25, 29, 39, 55, 57, 41, 40, 61, 79, 97, 132, 161, 133, 65, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 424, 448,
.305, 99, 11, 18, 25, 29, 39, 55, 57, 41, 40, 61, 79, 97, 132, 161, 133, 65, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 424, 449, 309, 113, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 425, 455, 331, 170, 141, 201, 255, 327, 433, 489, 432, 385, 483, 657, 836, 1076, 1296, 1200,
.705, 195, 11, 18, 25, 29, 39, 55, 57, 41, 40, 61, 79, 97, 132, 161, 133, 65, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 424, 449, 309, 113, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 425, 455, 331, 170, 141, 201, 255, 327, 433, 489, 432, 385, 483, 657, 836, 1076, 1296, 1201, 709, 209, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 425, 455, 331, 170, 141, 201, 255, 327, 433, 489, 432, 385, 483, 657, 836, 1076, 1297, 1207, 731, 266, 141, 201, 255, 327, 433, 489, 432, 385, 483, 657, 836, 1077, 1305, 1241, 832, 481, 483, 657, 837, 1087, 1355, 1410, 1249, 1253, 1623, ...
then the rows (omitting the first two terms of each row) converge to A151748.

David Applegate, The movie version [See A151725, variant]
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. A151725, A151726, A151735, A151748, A139250, A170879.

The first column gives A170881.

A151747 := proc(n) option remember; local i, j;
if (n <= 0) then
  0;
elif (n <= 3) then
  2*n-1;
else
   i := floor(log(n)/log(2));
   j := n - 2^i;
   if (j = 0) then (3*i+1)*2^(i-2)+1;
   elif (j = 1) then 3*2^(i-1)+3;
   elif (j = 2^i-1) then 2*procname(j)+procname(j+1)-1;
   else 2*procname(j)+procname(j+1);
   end if;
end if;
end proc;

a[n_] := a[n] = Module[{i, j}, Which[n <= 0, 0, n <= 3, 2n-1, True, i = Floor[Log2[n]]; j = n-2^i; Which[j == 0, (3i+1)*2^(i-2)+1, j == 1, 3*2^(i-1)+3, j == 2^i-1, 2a[j] + a[j+1] - 1,True, 2a[j] + a[j+1]]]];
Table[a[n], {n, 0, 67}] (* Jean-François Alcover, Aug 04 2022, from Maple code *)

A151746 G.f.: (1-2x-5x^2+4x^3)/((1-4x)*(1-x)^2).

Original entry on oeis.org

1, 4, 10, 32, 118, 460, 1826, 7288, 29134, 116516, 466042, 1864144, 7456550, 29826172, 119304658, 477218600, 1908874366, 7635497428, 30541989674, 122167958656, 488671834582, 1954687338284, 7818749353090, 31274997412312, 125099989649198, 500399958596740

Offset: 0

N. J. A. Sloane, Jun 16 2009

nonn

Subsequence of A151735 at powers of 2.

CoefficientList[Series[(1-2x-5x^2+4x^3)/((1-4x)(1-x)^2),{x,0,30}],x] (* Harvey P. Dale, Jan 13 2015 *)

a(n) = 14/9+2*n/3+4^(n+1)/9, n>0. [From R. J. Mathar, Jun 17 2009]

cp's OEIS Frontend

A151737 First differences of A151735 (starting at A151735(2)).

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A151725 Number of ON states after n generations of cellular automaton rule described by the rulestring B1/S012345678.

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A151747 Except for boundary cases (n <= 3, j = 0, 1, 2^i-1), satisfies a(n) = a(2^i+j) = 2 a(j) + a(j+1), where n = 2^i + j, 0 <= j < 2^i .

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A151746 G.f.: (1-2x-5x^2+4x^3)/((1-4x)*(1-x)^2).

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cp's OEIS Frontend

A151737 First differences of A151735 (starting at A151735(2)).

Views

Author

Keywords

Examples

Crossrefs

A151725 Number of ON states after n generations of cellular automaton rule described by the rulestring B1/S012345678.

Views

Author

Keywords

Comments

Links

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Programs

Mathematica

Formula

Extensions

A151747 Except for boundary cases (n <= 3, j = 0, 1, 2^i-1), satisfies a(n) = a(2^i+j) = 2 a(j) + a(j+1), where n = 2^i + j, 0 <= j < 2^i .

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Maple

Mathematica

A151746 G.f.: (1-2*x-5*x^2+4*x^3)/((1-4*x)*(1-x)^2).

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A151746 G.f.: (1-2x-5x^2+4x^3)/((1-4x)*(1-x)^2).