David Applegate - cp's OEIS frontend

A290867 Irregular triangle read by rows: the number of points that are the intersections of k semicircles in the configuration A290447(n).

0, 0, 0, 0, 1, 0, 5, 0, 15, 0, 35, 0, 70, 0, 123, 1, 0, 195, 5, 0, 285, 15, 0, 420, 25, 0, 586, 39, 2, 0, 818, 53, 4, 0, 1110, 73, 6, 0, 1451, 103, 10, 0, 1846, 142, 18, 0, 2361, 181, 26, 0, 2956, 234, 33, 2, 0, 3704, 287, 40, 4, 0, 4567, 348, 49, 8

Offset: 1

David Applegate, Aug 12 2017

Row lengths are A290726(n).
The first entry of each row is 0, because an intersection requires at least 2 lines.
The first row with 3 entries is for n=9, because that is the first configuration with a nontrivial intersection.
Row sums give A290447.

			Triangle begins:
  0;
  0;
  0;
  0,   1;
  0,   5;
  0,  15;
  0,  35;
  0,  70;
  0, 123,   1;
  0, 195,   5;
  0, 285,  15;
  0, 420,  25;
  0, 586,  39,   2;

David Applegate, Table of n, a(n) for n = 1..800
David Applegate, Triangular table T(n,k) for n = 1..100
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

Cf. A290447, A290726, A290865, A290866.

Sum_{k} T(n,k) * binomial(k,2) = binomial(n,4), because there are binomial(n,4) total pairs of semicircles, and an intersection of k consists of binomial(k,2) of those pairs.

A290865(n) = binomial(n,2) + Sum_{k} T(n,k) * (k-1).

A290866 a(n) = number of segments (edges) in the configuration A290447(n).

Original entry on oeis.org

0, 1, 3, 8, 20, 45, 91, 168, 285, 450, 670, 981, 1375, 1902, 2568, 3371, 4326, 5522, 6927, 8639, 10624, 12882, 15489, 18559, 22006, 25904, 30321, 35254, 40728, 46959, 53721, 61354, 69734, 78917, 89029, 100018, 111758, 124759, 138943

Offset: 1

David Applegate, Aug 12 2017

nonn

Only edges above the line are counted. Total edges = a(n) + n - 1.

David Applegate, Table of n, a(n) for n = 1..100

Cf. A290447, A290865, A290867.

a(n) = A290447(n) + A290865(n).

A290865 a(n) = number of regions in the configuration A290447(n).

Original entry on oeis.org

0, 1, 3, 7, 15, 30, 56, 98, 161, 250, 370, 536, 748, 1027, 1379, 1807, 2320, 2954, 3702, 4604, 5652, 6852, 8239, 9858, 11683, 13748, 16086, 18700, 21604, 24887, 28471, 32491, 36907, 41751, 47080, 52876, 59105, 65965, 73440, 81521, 90176

Offset: 1

David Applegate, Aug 12 2017

nonn

			With 3 points, there are 3 semicircles above the baseline, which bound a(3) = 3 regions. With 4 points, there are 6 semicircles, defining 7 regions (use the Halser webpage with n = 3 and 4). - _N. J. A. Sloane_, Aug 12 2017

David Applegate, Table of n, a(n) for n = 1..100
M. F. Hasler, Interactive web page for drawing the illustration for a(n).
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

Cf. A290447, A290866, A290867, A332723 (number of regions with k edges).

A266533 First differences of A266532.

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 9, 21, 3, 9, 9, 21, 9, 21, 21, 45, 3, 9, 9, 21, 9, 21, 21, 45, 9, 21, 21, 45, 21, 45, 45, 93, 3, 9, 9, 21, 9, 21, 21, 45, 9, 21, 21, 45, 21, 45, 45, 93, 9, 21, 21, 45, 21, 45, 45, 93, 21, 45, 45, 93, 45, 93, 93, 189, 3, 9, 9, 21, 9, 21, 21, 45, 9, 21, 21, 45, 21, 45, 45, 93

Offset: 1

David Applegate and Omar E. Pol, Jan 18 2016

Number of Y-toothpicks added at n-th stage in the structure of A266532.

A simplified version of A160121.

			Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
3;
3, 9;
3, 9, 9, 21;
3, 9, 9, 21, 9, 21, 21, 45;
3, 9, 9, 21, 9, 21, 21, 45, 9, 21, 21, 45, 21, 45, 45, 93;
...
Observation: at least the first 11 terms of the right border coincide with A068156.

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences

Cf. A000120, A011782, A038573, A068156. A139251, A147582, A160121, A151710, A160721, A266532, A267701.

a(1) = 1. It appears that a(n) = 3*A038573(n-1), n >= 2.

A266532 Total number of Y-toothpicks after n-th stage in the "outward" version of the cellular automaton of A160120.

Original entry on oeis.org

0, 1, 4, 7, 16, 19, 28, 37, 58, 61, 70, 79, 100, 109, 130, 151, 196, 199, 208, 217, 238, 247, 268, 289, 334, 343, 364, 385, 430, 451, 496, 541, 634, 637, 646, 655, 676, 685, 706, 727, 772, 781, 802, 823, 868, 889, 934, 979, 1072, 1081, 1102, 1123, 1168, 1189, 1234, 1279, 1372, 1393, 1438, 1483, 1576, 1621, 1714, 1807, 1996, 1999, 2008, 2017

Offset: 0

David Applegate and Omar E. Pol, Jan 18 2016

nonn

For the connection with A160720 (the "outward" version of the Ulam-Warburton cellular automaton A147562) see formula section and A267700.
A266533 (the first differences) gives the number of Y-toothpicks added to the structure at n-th stage.
First differs from A160120 at a(9).
First differs from A160715 at a(13).

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences

Cf. A139250, A147562, A159912, A160120, A160715, A160720, A266533, A267700, A267701.

Conjecture: a(n) = 1 + 3*(A160720(n) - 1)/4 = 1 + 3*A267700(n-1), n >= 1. This formula is correct! - N. J. A. Sloane, Jan 23 2016

a(n) = 1 + 3*(A159912(n) - n)/2, n >= 1. - Omar E. Pol, Jan 24 2016

A267191 Number of new cells turned ON at generation n in cellular automaton described in A267190.

Original entry on oeis.org

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

Offset: 1

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

nonn

First differences of A267190, which has much more information.

David Applegate, Table of n, a(n) for n = 1..260
David Applegate, The movie version

Cf. A267190.

Corrected by David Applegate, Jan 30 2016

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.

A261925 Nonpalindromes that are the sum of two nonzero palindromes of the same length.

Original entry on oeis.org

10, 12, 13, 14, 15, 16, 17, 18, 110, 132, 143, 154, 165, 176, 187, 198, 302, 312, 322, 332, 342, 352, 362, 372, 382, 403, 413, 423, 433, 443, 453, 463, 473, 483, 504, 514, 524, 534, 544, 554, 564, 574, 584, 605, 615, 625, 635, 645, 655, 665, 675, 685, 706, 716, 726

Offset: 1

David Applegate and N. J. A. Sloane, Sep 17 2015

Cf. A002113, A261921, A261924, etc.

A261924 Numbers that are the sum of two palindromes of the same length.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 198, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 302, 303, 312, 313, 322, 323, 332, 333, 342, 343, 352, 353, 362, 363, 372, 373, 382, 383, 393

Offset: 1

David Applegate and N. J. A. Sloane, Sep 17 2015

Theorem: For a fixed value of d, adding two palindromes of length d in all possible ways produces 19 distinct sums if d=1, and 17*19^floor((d-1)/2) distinct sums if d>1. (The number of palindromes with d digits is 10 if d = 1, otherwise 9*10^floor((d-1)/2).) - N. J. A. Sloane, Dec 06 2015

N. J. A. Sloane, Table of n, a(n) for n = 1..12956

Cf. A002113, A261921, A261925, etc.

Modified to include the zero palindrome. - N. J. A. Sloane, Dec 06 2015

A261921 Nonpalindromes which are the sum of two palindromes whose lengths differ by 1.

Original entry on oeis.org

12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 47, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81, 82, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94, 95, 96, 97, 100, 101, 102, 103, 104, 105, 106, 107, 108, 112, 122, 123, 132

Offset: 1

David Applegate and N. J. A. Sloane, Sep 15 2015

More than the usual number of terms are displayed in order to show that this is different from A261907.

			12=11+1 and 100=99+1 are members.
10 is not a member since it is only the sum of two palindromes (5+5) whose lengths are equal.

Cf. A002113. A subsequence of A261907. A261920 shows the differences.

cp's OEIS Frontend

User: David Applegate

A290867 Irregular triangle read by rows: the number of points that are the intersections of k semicircles in the configuration A290447(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A290866 a(n) = number of segments (edges) in the configuration A290447(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A290865 a(n) = number of regions in the configuration A290447(n).

Views

Author

Keywords

Examples

Links

Crossrefs

A266533 First differences of A266532.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A266532 Total number of Y-toothpicks after n-th stage in the "outward" version of the cellular automaton of A160120.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A267191 Number of new cells turned ON at generation n in cellular automaton described in A267190.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

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

A261925 Nonpalindromes that are the sum of two nonzero palindromes of the same length.

Views

Author

Keywords

Crossrefs

A261924 Numbers that are the sum of two palindromes of the same length.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A261921 Nonpalindromes which are the sum of two palindromes whose lengths differ by 1.

Views

Author

Keywords

Comments

Examples