A170896 -id:A170896 - cp's OEIS frontend

A006447 Consider a 2-D cellular automaton generated by the Schrandt-Ulam rule of A170896, but confined to a semi-infinite strip of width n, starting with one ON cell at the top left corner; a(n) is the period of the resulting structure.

1, 2, 3, 5, 5, 8, 13, 13, 13, 26, 13, 91, 13, 106, 106, 75, 93, 62, 80, 132, 337, 416, 62, 62, 62, 271, 34, 155, 525, 548, 1084, 115, 62, 558, 62, 1500, 2922, 124, 2958, 3374, 2323, 4183, 8073, 7925, 744, 2298, 434, 6700, 310, 23796, 12732, 26405

Offset: 1

N. J. A. Sloane

nonn

Schrandt and Ulam remark that there seems to be no simple relation between n and a(n).

The original report included two further terms, but they were omitted from the published version, so are presumably unreliable.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. T. Butler, Letter to N. J. A. Sloane, Jun. 1975.
R. G. Schrandt and S. M. Ulam, On recursively defined geometric objects and patterns of growth, Los Alamos Scientific Laboratory, Report LA-3762, Aug 16 1967; published in A. W. Burks, editor, Essays on Cellular Automata. Univ. Ill. Press, 1970, pp. 238ff. [Link supplied by Laurinda J. Alcorn, Jan 09 2010]
Index entries for sequences related to cellular automata

Entry revised by N. J. A. Sloane, Jan 09 2010

More terms from Sean A. Irvine, Apr 13 2017

A170897 Number of new cells turned ON at generation n in cellular automaton described in A170896.

Original entry on oeis.org

0, 1, 4, 4, 4, 12, 4, 12, 12, 12, 20, 12, 20, 28, 12, 12, 12, 20, 28, 20, 36, 36, 44, 44, 36, 52, 52, 28, 28, 28, 36, 44, 44, 68, 68, 76, 76, 60, 68, 68, 68, 68, 60, 36, 68, 68, 84, 100, 76, 76, 92, 68, 84, 92, 100, 140, 100, 132, 116, 100, 84, 76, 108, 140, 108, 100, 100, 132, 92, 156, 172, 124, 76, 108, 108, 124, 164, 140, 148, 132, 116, 108, 172, 156, 132, 108, 140, 148, 196

Offset: 0

N. J. A. Sloane, Jan 09 2010

nonn

First differences of A170896. See that entry for much more information.

David Applegate, Table of n, a(n) for n = 0..260 (a(145) corrected by Sean A. Irvine)
David Applegate, The movie version
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.

Cf. A170896.

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

A266536 Total number of ON cells after n-th stage in a 90-degree sector of the cellular automaton of A170896.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 10, 13, 16, 21, 24, 29, 36, 39, 42, 45, 50, 57, 62, 71, 80, 91, 102, 111, 124, 137, 144, 151, 158, 167, 178, 189, 206, 223, 242, 261, 276, 293, 310, 327, 344, 359, 368, 385, 402, 423, 448, 467, 486, 509, 526, 547, 570, 595, 630, 655, 688, 717, 742, 763, 782, 809, 844, 871, 896, 921, 954, 977, 1016, 1059

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 A170896.
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 both A161336 (snowflake tree) and A266534 at a(13).

Cf. A161336, A169779, A170896, A266534, A267700.

a(n) = (A170896(n+1) - 1)/4.

A151895 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, 185, 213, 233, 261, 297, 333, 385, 429, 481, 533, 545, 573, 601, 629, 673, 717, 761, 837, 905, 989, 1033, 1085, 1145, 1197, 1257, 1309, 1337, 1397, 1457, 1525, 1625, 1669

Offset: 0

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

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 considered (that is, satisfied a)) in any previous generation, and
c) Q's outer squares are not prospective squares of the (n+1)st generation satisfying a).
Originally constructed in an attempt to explain the Holladay-Ulam CA shown in Fig. 2 of the 1962 Ulam article. However, as explained on page 222 of that article, the actual rule for that CA (see A151906, A151907) is different from ours.
A170896 and A267190 are also closely related cellular automata.
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..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.], which is also available at arXiv:1004.3036v2
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, Illustration of initial terms (concentrating on a 90-degree sector)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
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]

See A170896, A170897 for the original Schrandt-Ulam version.

Cf. A151896 (the first differences), A139250, A151905, A151906, A151907, A267190, A267191.

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

A170802 a(n) = n^10*(n^10 + 1)/2.

Original entry on oeis.org

0, 1, 524800, 1743421725, 549756338176, 47683720703125, 1828079250264576, 39896133290043625, 576460752840294400, 6078832731271856601, 50000000005000000000, 336374997479248716901, 1916879996254696243200

Offset: 0

N. J. A. Sloane, Dec 11 2009

By definition, all terms are triangular numbers. - Harvey P. Dale, Aug 12 2012

Number of unoriented rows of length 20 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=524800, there are 2^20=1048576 oriented arrangements of two colors. Of these, 2^10=1024 are achiral. That leaves (1048576-1024)/2=523776 chiral pairs. Adding achiral and chiral, we get 524800. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Row 20 of A277504.
Cf. A010808 (oriented), A008454 (achiral).
Sequences of the form n^10*(n^m + 1)/2: A170793 (m=1), A170794 (m=2), A170795 (m=3), A170896 (m=4), A170797 (m=5), A170798 (m=6), A170799 (m=7), A170800 (m=8), A170801 (m=9), this sequence (m=10).

List([0..30], n -> n^10*(n^10+1)/2); # G. C. Greubel, Nov 15 2018

[n^10*(n^10+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

seq(n^10*(n^10 +1)/2, n=0..20); # G. C. Greubel, Oct 11 2019

n10[n_]:=Module[{c=n^10},(c(c+1))/2];Array[n10,15,0] (* Harvey P. Dale, Jul 17 2012 *)

vector(30, n, n--; n^10*(n^10+1)/2) \\ G. C. Greubel, Nov 15 2018

for n in range(0,20): print(int(n**10*(n**10 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

[n^10*(n^10+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010808(n) + A008454(n)) / 2 = (n^20 + n^10) / 2.
G.f.: (Sum_{j=1..20} S2(20,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..10} S2(10,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..19} A145882(20,k) * x^k / (1-x)^21.
E.g.f.: (Sum_{k=1..20} S2(20,k)*x^k + Sum_{k=1..10} S2(10,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>20, a(n) = Sum_{j=1..21} -binomial(j-22,j) * a(n-j). (End)

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

A006447 Consider a 2-D cellular automaton generated by the Schrandt-Ulam rule of A170896, but confined to a semi-infinite strip of width n, starting with one ON cell at the top left corner; a(n) is the period of the resulting structure.

Views

Author

Keywords

Comments

References

Links

Extensions

A170897 Number of new cells turned ON at generation n in cellular automaton described in A170896.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A266536 Total number of ON cells after n-th stage in a 90-degree sector of the cellular automaton of A170896.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A151895 Number of ON cells after n generations of the cellular automaton on the square grid that is described in the Comments.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A170802 a(n) = n^10*(n^10 + 1)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Python

Sage

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula