A151747 -id:A151747 - cp's OEIS frontend

A151748 If the first two terms of the rows of the triangle in A151747 are omitted, this is what the rows converge to.

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

Offset: 0

N. J. A. Sloane, Jun 16 2009, Jun 17 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

A170879 Partial sums of A151747.

Original entry on oeis.org

0, 1, 4, 9, 17, 26, 37, 54, 75, 90, 101, 119, 144, 173, 212, 266, 319, 346, 357, 375, 400, 429, 468, 523, 580, 621, 661, 722, 801, 898, 1030, 1190, 1319, 1370, 1381, 1399, 1424, 1453, 1492, 1547, 1604, 1645, 1685, 1746, 1825, 1922, 2054, 2215, 2348, 2413, 2453

Offset: 0

N. J. A. Sloane, Jan 07 2010

nonn

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. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

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

A151728 A151727/4.

Original entry on oeis.org

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, 7, 15, 21, 29, 29, 49, 61, 53, 29, 51, 71, 87, 107, 157, 165, 101, 29, 51, 71, 87, 107, 159, 175, 135, 109, 173, 229, 281, 371, 477

Offset: 0

N. J. A. Sloane, Jun 14 2009

nonn

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. A151727, A151737, A151747, A170880. Equals 2*A151729 + 1.

A170881 a(0)=0; thereafter a(n) = (3n+1)2^(n-2)+1.

Original entry on oeis.org

0, 1, 3, 8, 21, 53, 129, 305, 705, 1601, 3585, 7937, 17409, 37889, 81921, 176129, 376833, 802817, 1703937, 3604481, 7602177, 15990785, 33554433, 70254593, 146800641, 306184193, 637534209, 1325400065, 2751463425, 5704253441, 11811160065, 24427626497, 50465865729

Offset: 0

N. J. A. Sloane, Jan 07 2010

Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Essentially the first column of the triangular array in A151747.

Partial sums of A098156.

Join[{0,1},Table[(3n+1)2^(n-2)+1,{n,40}]] (* Harvey P. Dale, Nov 26 2023 *)

From Chai Wah Wu, Apr 15 2025: (Start)
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3) for n > 4.
G.f.: x*(-x^3 - x^2 + 2*x - 1)/((x - 1)*(2*x - 1)^2). (End)
E.g.f.: (4*exp(x) - 3 + exp(2*x)*(3*x - 1))/4. - Stefano Spezia, Apr 15 2025

Zero prepended by Harvey P. Dale, Nov 26 2023

cp's OEIS Frontend

A151748 If the first two terms of the rows of the triangle in A151747 are omitted, this is what the rows converge to.

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A170879 Partial sums of A151747.

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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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A151728 A151727/4.

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A170881 a(0)=0; thereafter a(n) = (3n+1)2^(n-2)+1.

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Formula

Extensions

cp's OEIS Frontend

A151748 If the first two terms of the rows of the triangle in A151747 are omitted, this is what the rows converge to.

Views

Author

Keywords

Links

A170879 Partial sums of A151747.

Views

Author

Keywords

Links

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A151728 A151727/4.

Views

Author

Keywords

Links

Crossrefs

A170881 a(0)=0; thereafter a(n) = (3*n+1)*2^(n-2)+1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A170881 a(0)=0; thereafter a(n) = (3n+1)2^(n-2)+1.