A151895 -id:A151895 - cp's OEIS frontend

A151896 Number of new cells turned ON at generation n in cellular automaton described in A151895.

0, 1, 4, 4, 4, 12, 4, 12, 12, 12, 20, 12, 20, 28, 4, 12, 12, 12, 28, 20, 28, 36, 36, 52, 44, 52, 52, 12, 28, 28, 28, 44, 44, 44, 76, 68, 84, 44, 52, 60, 52, 60, 52, 28, 60, 60, 68, 100, 44, 68, 84, 60, 92, 84, 76, 116, 84, 100, 92, 60, 60, 76, 92, 132, 100, 84

Offset: 0

David Applegate and N. J. A. Sloane, Jul 30 2009

nonn

First differences of A151895. See that entry for much more information.

David Applegate, Table of n, a(n) for n = 0..250
David Applegate, The movie version
David Applegate, Illustration of first 10 generations
David Applegate, Illustration of first 20 generations
David Applegate, Illustration of first 32 generations
David Applegate, Illustration of first 64 generations
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, Illustration of initial terms (concentrating on a 90-degree sector)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A151895, A139250.

We do not know of a recurrence or generating function.

Revised by David Applegate and N. J. A. Sloane, Jan 21 2016

A266534 Total number of ON cells after n-th stage in a 90-degree sector of the cellular automaton of A151895.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 10, 13, 16, 21, 24, 29, 36, 37, 40, 43, 46, 53, 58, 65, 74, 83, 96, 107, 120, 133, 136, 143, 150, 157, 168, 179, 190, 209, 226, 247, 258, 271, 286, 299, 314, 327, 334, 349, 364, 381, 406, 417, 434, 455, 470, 493, 514, 533, 562, 583, 608, 631, 646, 661, 680, 703, 736, 761, 782, 807, 836, 857, 892, 927

Offset: 0

Omar E. Pol, Jan 12 2016

nonn

The structure looks like a tree which arises from one of the four spokes of the structure of the cellular automaton of A151895.
a(n) is the total number of ON cells after n-th stage.
For n >> 1 the structure looks like a square which is rotated 45 degrees.
First differs from A161336 (snowflake tree) at a(16).
First differs from A266536 at a(13). - Omar E. Pol, Apr 02 2016

Cf. A151895, A161330, A161336, A169779, A266536, A267700.

a(n) = (A151895(n+1) - 1)/4.

More terms from Omar E. Pol, Apr 02 2016

A151907 Partial sums of A151906.

Original entry on oeis.org

0, 1, 5, 9, 13, 25, 29, 33, 45, 57, 69, 105, 109, 113, 125, 137, 149, 185, 197, 209, 245, 281, 317, 425, 429, 433, 445, 457, 469, 505, 517, 529, 565, 601, 637, 745, 757, 769, 805, 841, 877, 985, 1021, 1057, 1165, 1273, 1381, 1705, 1709, 1713, 1725, 1737, 1749, 1785, 1797

Offset: 0

David Applegate and N. J. A. Sloane, Jul 31 2009, Aug 03 2009

nonn

a(n) is the total number of ON cells after n generations in the Holladay-Ulam CA shown in Fig. 2 and Example 2 of the Ulam article.
The definition of this CA given by Ulam on pp. 216, 222 is complicated (and incomplete). However, the same structure can be obtained as follows. Take the CA of A147562 but replace each square by a Maltese cross, a cluster of five adjacent squares:
..X..
.XXX.
..X..
We guess that this was how Holladay and Ulam originally constructed the CA.
This construction corresponds to the fact that the three trisections of the difference sequence A151906 are essentially A147582, A147582 and 3*A147582 respectively.

S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.

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
N. J. A. Sloane, Illustration of initial terms (annotated copy of figure on p. 222 of Ulam)

Cf. A151904, A151905, A151906, A139250, A151895, A151896.

The definition in terms of A151906 provides an explicit formula for a(n).

A151906 a(0) = 0, a(1) = 1; for n>1, a(n) = 8*A151905(n) + 4.

Original entry on oeis.org

0, 1, 4, 4, 4, 12, 4, 4, 12, 12, 12, 36, 4, 4, 12, 12, 12, 36, 12, 12, 36, 36, 36, 108, 4, 4, 12, 12, 12, 36, 12, 12, 36, 36, 36, 108, 12, 12, 36, 36, 36, 108, 36, 36, 108, 108, 108, 324, 4, 4, 12, 12, 12, 36, 12, 12, 36, 36, 36, 108, 12, 12, 36, 36, 36, 108, 36, 36, 108, 108, 108

Offset: 0

David Applegate and N. J. A. Sloane, Jul 31 2009, Aug 03 2009

Consider the Holladay-Ulam CA shown in Fig. 2 and Example 2 of the Ulam article. Then a(n) is the number of cells turned ON in generation n.

			From _Omar E. Pol_, Apr 02 2018: (Start)
Note that this sequence also can be written as an irregular triangle read by rows in which the row lengths are the terms of A011782 multiplied by 3, as shown below:
0,1, 4;
4,4,12;
4,4,12,12,12,36;
4,4,12,12,12,36,12,12,36,36,36,108;
4,4,12,12,12,36,12,12,36,36,36,108,12,12,36,36,36,108,36,36,108,108,108,324;
4,4,12,12,12,36,12,12,36,36,36,108,12,12,36,36,36,108,36,36,108,108,108,... (End)

S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.

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
N. J. A. Sloane, Illustration of initial terms (annotated copy of figure on p. 222 of Ulam)

Cf. A151904, A151905, A151907, A139250, A151895, A151896.

f := proc(n) local j; j:=n mod 6; if (j<=1) then 0 elif (j<=4) then 1 else 2; fi; end;
wt := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end;
A151904 := proc(n) local k,j; k:=floor(n/6); j:=n-6*k; (3^(wt(k)+f(j))-1)/2; end;
A151905 := proc (n) local k,j;
if (n=0) then 0;
elif (n=1) then 1;
elif (n=2) then 0;
else k:=floor( log(n/3)/log(2) ); j:=n-3*2^k; A151904(j); fi;
end;
A151906 := proc(n);
if (n=0) then 0;
elif (n=1) then 1;
else 8*A151905(n) + 4;
fi;
end;

wt[n_] := DigitCount[n, 2, 1];
f[n_] := {0, 0, 1, 1, 1, 2}[[Mod[n, 6] + 1]];
A151902[n_] := wt[Floor[n/6]] + f[n - 6 Floor[n/6]];
A151904[n_] := (3^A151902[n] - 1)/2;
A151905[n_] := Module[{k, j}, Switch[n, 0, 0, 1, 1, 2, 0, _, k = Floor[Log2[n/3]]; j = n - 3*2^k; A151904[j]]];
a[n_] := Switch[n, 0, 0, 1, 1, _, 8 A151905[n] + 4];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 16 2023, after Maple code *)

The three trisections are essentially A147582, A147582 and 3*A147582 respectively. More precisely, For t >= 1, a(3t) = a(3t+1) = A147582(t+1) = 4*3^(wt(t)-1), a(3t+2) = 4*A147582(t+1) = 4*3^wt(t). See A151907 for explanation.

A170896 Number of ON cells after n generations of the Schrandt-Ulam cellular automaton on the square grid that is described in the Comments.

Original entry on oeis.org

0, 1, 5, 9, 13, 25, 29, 41, 53, 65, 85, 97, 117, 145, 157, 169, 181, 201, 229, 249, 285, 321, 365, 409, 445, 497, 549, 577, 605, 633, 669, 713, 757, 825, 893, 969, 1045, 1105, 1173, 1241, 1309, 1377, 1437, 1473, 1541, 1609, 1693, 1793, 1869, 1945, 2037, 2105, 2189, 2281, 2381, 2521, 2621, 2753, 2869, 2969, 3053, 3129, 3237, 3377, 3485, 3585, 3685, 3817, 3909

Offset: 0

N. J. A. Sloane, Jan 09 2010

nonn

The cells are the squares of the standard square grid.
Cells are either OFF or ON, once they are ON they stay ON, and we begin in generation 1 with 1 ON cell.
Each cell has 4 neighbors, those that it shares an edge with. Cells that are ON at generation n all try simultaneously to turn ON all their neighbors that are OFF. They can only do this at this point in time; afterwards they go to sleep (but stay ON).
A square Q is turned ON at generation n+1 if:
a) Q shares an edge with one and only one square P (say) that was turned ON at generation n (in which case the two squares which intersect Q only in a vertex not on that edge are called Q's "outer squares"), and
b) Q's outer squares were not turned ON in any previous generation.
c) In addition, of this set of prospective squares of the (n+1)th generation satisfying the previous condition, we eliminate all squares which are outer squares of other prospective squares.
A151895, A151906, and A267190 are closely related cellular automata.

D. 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.

David Applegate, Table of n, a(n) for n = 0..260 (corrected by Sean A. Irvine)
David Applegate, The movie version
David Applegate, After 20 generations, illustrating a(20)=285 (with the A170897(20)=36 newly created cells shown in blue)
David Applegate, After 26 generations, illustrating a(26)=549 (with the A170897(26)=52 newly created cells shown in blue)
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.]
Sean A. Irvine, Java program (github)
R. G. Schrandt and S. M. Ulam, On recursively defined geometric objects and patterns of growth [Link supplied by Laurinda J. Alcorn, Jan 09 2010.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021
S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 [Annotated scanned copy]

Cf. A139250, A170897 (first differences), A151895, A151896, A151906, A267190.

We do not know of a recurrence or generating function.

Entry (including definition) revised by David Applegate and N. J. A. Sloane, Jan 21 2016

A151905 a(0) = a(2) = 0, a(1) = 1; for n >= 3, n = 32^k+j, 0 <= j < 32^k, a(n) = A151904(j).

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 4, 0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 4, 4, 13, 13, 13, 40, 13

Offset: 0

N. J. A. Sloane, Jul 31 2009

Consider the Holladay-Ulam CA shown in Fig. 2 and Example 2 of the Ulam article. Then a(n) is the number of cells turned ON in generation n in a 45-degree sector that are not on the main stem.

			If written as a triangle:
0,
1, 0,
0, 0, 1,
0, 0, 1, 1, 1, 4,
0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13,
0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40
0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 4, 4, 13, 13, 13, 40, 13, 13, 40, 40, 40, 121,
...
then the rows converge to A151904.

S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.

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
N. J. A. Sloane, Illustration of initial terms (annotated copy of figure on p. 222 of Ulam)

Cf. A151904, A151906, A151907, A139250, A151895, A151896.

f := proc(n) local j; j:=n mod 6; if (j<=1) then 0 elif (j<=4) then 1 else 2; fi; end;
wt := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end;
A151904 := proc(n) local k,j; k:=floor(n/6); j:=n-6*k; (3^(wt(k)+f(j))-1)/2; end;
A151905 := proc (n) local k,j;
if (n=0) then 0;
elif (n=1) then 1;
elif (n=2) then 0;
else k:=floor( log(n/3)/log(2) ); j:=n-3*2^k; A151904(j); fi;
end;

wt[n_] := DigitCount[n, 2, 1];
f[n_] := {0, 0, 1, 1, 1, 2}[[Mod[n, 6] + 1]];
A151902[n_] := wt[Floor[n/6]] + f[n - 6 Floor[n/6]];
A151904[n_] := (3^A151902[n] - 1)/2;
a[n_] := Module[{k, j}, Switch[n, 0, 0, 1, 1, 2, 0, _, k = Floor[Log2[n/3]]; j = n - 3*2^k; A151904[j]]];
Table[a[n], {n, 0, 90}] (* Jean-François Alcover, Feb 16 2023, after Maple code *)

A267190 Number of ON cells after n generations of the cellular automaton on the square grid that is described in the Comments.

Original entry on oeis.org

0, 1, 5, 9, 13, 25, 29, 41, 53, 65, 85, 97, 117, 145, 149, 161, 173, 193, 221, 241, 277, 313, 357, 401, 437, 489, 541, 553, 581, 609, 645, 689, 733, 801, 869, 945, 1021, 1081, 1149, 1217, 1277, 1345, 1397, 1433, 1501, 1569, 1653, 1753, 1829, 1905, 1997, 2057, 2141, 2225, 2317, 2449, 2549, 2681, 2797, 2889, 2965, 3041, 3149, 3289

Offset: 0

David Applegate and N. J. A. Sloane, Jan 21 2016

nonn

The cells are the squares of the standard square grid.
Cells are either OFF or ON, once they are ON they stay ON, and we begin in generation 1 with 1 ON cell.
Each cell has 4 neighbors, those that it shares an edge with. Cells that are ON at generation n all try simultaneously to turn ON all their neighbors that are OFF. They can only do this at this point in time; afterwards they go to sleep (but stay ON).
A square Q is turned ON at generation n+1 if:
a) Q shares an edge with one and only one square P (say) that was turned ON at generation n (in which case the two squares which intersect Q only in a vertex not on that edge are called Q's “outer squares”), and
b) Q's outer squares were not turned ON in any previous generation, and
c) Q's outer squares are not prospective squares of the (n+1)st generation satisfying a).
A151895, A151906, and A170896 are closely related cellular automata.
The key difference between this and A170896 is that if we have two squares Q1 and Q2, both satisfying a), and that are each an outer square of the other, where Q1 satisfies b), but Q2 does not, then for A170896 Q1 is accepted, but for this sequence Q1 is eliminated.  This first happens at n=14, when, for example, A170896 turns (8,3) ON but A267190 doesn't (because (9,2) fails to satisfy b) because (8,1) is ON). - David Applegate, Jan 30 2016
A151895 and A267190 first differ at n=17, when A267190 turns (12,2) ON even though its outer square (11,1) was considered (not turned ON) in a previous generation. - David Applegate, Jan 30 2016

D. 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

David Applegate, Table of n, a(n) for n = 0..260
David Applegate, The movie version
David Applegate, After 16 generations, illustrating a(16)=173 (with the A267191(16)=12 newly created cells shown in blue)
David Applegate, After 36 generations, illustrating a(36)=1021 (with the A267191(36)=76 newly created cells shown in blue) 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.], which is also available at arXiv:1004.3036v2
S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 [Annotated scanned copy]
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Cf. A267191 (first differences), A151895, A151906, A170896.

A293392 Total number of ON cells after n-th stage in a 90-degree sector of the cellular automaton of A267190.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 10, 13, 16, 21, 24, 29, 36, 37, 40, 43, 48, 55, 60, 69, 78, 89, 100, 109, 122, 135, 138, 145, 152, 161, 172, 183, 200, 217, 236, 255, 270, 287, 304, 319, 336, 349, 358, 375, 392, 413, 438, 457, 476, 499, 514, 535, 556, 579, 612, 637, 670, 699, 722, 741, 760, 787, 822, 847, 872, 897, 930, 953, 992

Offset: 0

Omar E. Pol, Oct 08 2017

nonn

The structure looks like a tree which arises from one of the four spokes of the structure of the cellular automaton of A267190.
a(n) is the total number of ON cells after n-th stage.
For n >> 1 the structure looks like a square which is rotated 45 degrees.
First differs from A161336 at a(17), where A161336 is a version of A161330 (the snowflake cellular automaton).
First differs from A266534 at a(16), where A266534 is a version of A151895.
First differs from A266536 at a(13), where A266536 is a version of A170896 (the Schrandt-Ulam cellular automaton).
From Omar E. Pol, Oct 16 2017: (Start)
The graph of both A266536 and this sequence are very similar.
For n >> 1, it appears that A266534(n) < A161336(n) < a(n) < A266536(n).
The graphs of these four sequences are similar, and the behavior looks like percolation.
It appears that there are no recurrences in these four sequences. Thus it appears that there are no recurrences in A151895, A161330, A267190 and A170896. (End)

Cf. A151895, A161330, A161336, A170896, A266534, A266536, A267190, A267700.

a(n) = (A267190(n+1) - 1)/4.

cp's OEIS Frontend

A151896 Number of new cells turned ON at generation n in cellular automaton described in A151895.

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A266534 Total number of ON cells after n-th stage in a 90-degree sector of the cellular automaton of A151895.

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A151907 Partial sums of A151906.

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A151906 a(0) = 0, a(1) = 1; for n>1, a(n) = 8*A151905(n) + 4.

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A170896 Number of ON cells after n generations of the Schrandt-Ulam cellular automaton on the square grid that is described in the Comments.

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A151905 a(0) = a(2) = 0, a(1) = 1; for n >= 3, n = 3*2^k+j, 0 <= j < 3*2^k, a(n) = A151904(j).

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A267190 Number of ON cells after n generations of the cellular automaton on the square grid that is described in the Comments.

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Formula

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A293392 Total number of ON cells after n-th stage in a 90-degree sector of the cellular automaton of A267190.

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Author

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A151905 a(0) = a(2) = 0, a(1) = 1; for n >= 3, n = 32^k+j, 0 <= j < 32^k, a(n) = A151904(j).