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# Catalog of Toothpick and CA Sequences in OEIS

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# Catalog of Toothpick and CA Sequences in OEIS

## Contents

- The On-Line Encyclopedia of Integer Sequences (or OEIS) contains a large number of sequences connected with the "toothpick problem", associated cellular automata, and related recurrences and generating functions.
- The following list provides a searchable index to these sequences.
- This is a companion to the paper
**The Toothpick Sequence and Other Sequences from Cellular Automata**by David Applegate, Omar E. Pol and N. J. A. Sloane. - It was created by N. J. A. Sloane on Jan 10 2010, and unsigned entries are due to him.
- If you make any changes, please sign them with your name and date.

## Abbreviations

- CA = cellular automaton
- g.f. = generating function
- nbrs = neighbors

## The Sequences

- Toothpick sequence, original version: A139250, A139251
- Toothpick sequence, original version: limiting form of rows: A147646, A151688, A152980, A170927
- Toothpick sequence, original version: see also (1): A139252, A139253, A139254, A139255, A139559, A139560, A147614, A151567, A152768, A152968, A152973, A152978, A152979, A152998, A152999
- Toothpick sequence, original version: see also (2): A153000, A153002, A153003, A153004, A153005, A153838, A159786, A159787, A159788, A159789, A159799, A159790, A159791, A159792
- Toothpick sequence, original version: see also (3): A160124, A160125, A160126, A160127, A160128, A160158, A160159, A160162, A160163, A160164, A160165
- Toothpick sequence, original version: see also (4): A160704, A160424, A160426, A160427, A160730, A160731, A160732, A160733
- Toothpick sequence, original version: see also (5): A160734, A160735, A160762, A162625, A162627, A162793, A162794, A162795, A162796, A162797, A163094
- Toothpick sequence, original version: see also (6): A168002, A168112, A168113, A168114, A168115
- Toothpick sequence, version where state alternates: A151885, A151886, A151887, A151888
- Toothpick sequence, another version: A160410, A160412, A160414, A160416, A160430, A161411, A161415, A161417
- Toothpick sequence, version with longer toothpicks: A160420, A160421, A160422, A160423
- Toothpick sequence, in Fibonacci spiral: A160800, A160801, A160802, A160803, A160808, A160809

- Corner sequence: A152980, A153006
- Corner sequence, see also: A159795, A153001, A159801, A153007, A153009, A159785, A159793, A159794, A159795, A159796, A159797, A160166, A162777, A162779, A171170, A171171, A168131, A170926, A170929
- Corner sequence, another version: A160406, A160407, A160418, A160718, A160719, A160736, A160738, A160740
- Corner sequence, another version: see also A170884, A170884, A170886, A170887, A170888, A170889, A170890, A170891, A170892, A170893, A170894, A170895
- Corner sequence, 3-D version: A160408, A160409, A160419, A161210, A161211, A160728, A160729, A161212, A161214, A161216

Contribution from Omar E. Pol, Dec 23 2012 (Start):

- D-toothpick sequence: A194270, A194271, A194276, A194277, A194278, A194432, A194433, A194434, A194435, A194440, A194441, A194442, A194443, A194444, A194445
- D-toothpick sequence, corner version: A194692, A194693, A194694, A194695, A194696, A194697

(End)

- E-toothpick (or snowflake) sequence: A161328, A161329, A161330, A161331, A161332, A161333, A161334, A161335, A161336

Contribution from Omar E. Pol, Dec 23 2012 (Start):

(End)

Contribution from Omar E. Pol, Dec 23 2012 (Start):

(End)

Contribution from Omar E. Pol, Dec 23 2012 (Start):

- I-toothpick sequence: A160164

(End)

Contribution from Omar E. Pol, Dec 23 2012 (Start):

(End)

- Leftist toothpicks: A151566, A151565
- Leftist toothpicks: see also: Gould's sequence A001316
- Leftist toothpicks: see also: A160742, A160744, A160745, A160746, A175098, A175099
- Pascal triangle mod 2 (or Sierpinski triangle): A047999

Contribution from Omar E. Pol, Dec 23 2012 (Start):

(End)

- T-toothpick sequence: A160172, A160173
- T-toothpick sequence, corner version: A160714, A160716, A160724, A160726

- V-toothpick sequence: A161206, A161207
- V-toothpick sequence, another version: A161420, A161421, A161423
- V-toothpick sequence, corner version: A161412, A161413

- Y-toothpick sequence: A160120, A160121
- Y-toothpick sequence, another version: A160715, A151710
- Y-toothpick sequence: see also (1): A160122, A160123, A160157, A160167, A160425, A160789, A161418, A161426, A161427, A161429, A161430
- Y-toothpick sequence: see also (2): A161828, A161829, A161830, A161831, A161832, A161833, A161834, A161836, A161837, A161838, A161910

- Ulam-Warburton CA: A147562, A147582
- Ulam-Warburton CA: see also A079314, A079315, A079316, A079317, A151921, A079318
- Holladay-Ulam CA based on Maltese crosses: A151904, A151905, A151906, A151907
- Schrandt-Ulam CA: A170896, A170895; see also A151895, A151896

- Z^2 with 4 nbrs: A147562, A147582
- Z^2 with 4 nbrs, alternating states: A072272, A170878
- Z^2 with 4 nbrs: see also: A160720, A160721, A151895, A151896
- CAs based on Sierpinski triangles: A160722, A160723, A160720, A160721

- Z^2 with 8 nbrs: A151725, A151726, A151747, A151748, A151728, A170879, A170880
- Z^2 with 8 nbrs: other versions: A160117, A160118

- Hexagonal lattice with 6 nbrs (cells are hexagons): A151723, A151724, A170905, A170906, A170907
- Triangle-based CA: A161644, A161645, A170882, A170883

- 3-D toothpick structures: A160160, A160161, A160428, A170875, A170876, A170884, A170885
- 3-D toothpick structures: see also: A160430, A162798, A162799

- Recurrences a(2^i+j) = C*a(j) + D*a(j+1), a(0) = A, a(1) = B:
- (A B C D)
- (0 1 1 1): A118977, A163267
- (1 0 1 1): A151702
- (1 1 1 1): A151570
- (1 2 1 1): A151571
- (0 1 1 2): A151572
- (1 0 1 2): A151703
- (1 1 1 2): A151573
- (1 2 1 2): A151574
- (0 1 2 1): A160552, A151548 (row limit), A160570; see also A162956, A162958, A170903
- (1 0 2 1): A151704
- (1 1 2 1): A151568
- (1 2 2 1): A151569
- (0 1 2 2): A151705
- (1 0 2 2): A151706
- (1 1 2 2): A151707
- (1 2 2 2): A151708

- Others of this type: A162956, A170854, A170855, A170856, A170857, A170858, A170859, A170860, A170861, A170862, A170863, A170864, A170865, A170866, A170867, A170868, A170869, A170870, A170871, A170872

- Generating functions of the form Prod_{k>=c} (1 + a*x^(2^k-1) + b*x^2^k)):
- Others of this type:, A170838, A170839, A170840, A170841, A170842, A170843, A170844, A170845, A170846, A170847, A170848, A170849, A170850, A170851, A170852

- G.f.: (1+x) * Prod_{ n >= 1} (1 + x^(2^n-1) + x^(2^n)); A151553
- G.f.: (1+2x) * Prod_{ n >= 1} (1 + x^(2^n-1) + x^(2^n)): A151554
- G.f.: (1+2x) * Prod_{ n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)): A151555
- G.f.: (1+3x) * Prod_{ n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)): A151551

- Generating functions of the form Prod_{k>=0} (1 + a*x^(b^k)): for the following values of

- Sequences related to Wolfram's Rule 22, Rule 28, Rule 30, etc: see
**Index to sequences in the OEIS related to cellular automata**