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Catalog of toothpick and CA sequences in OEIS

# Catalog of Toothpick and Cellular Automata Sequences in OEIS

## N. J. A. Sloane

### Email: njasloane@gmail.com

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, as they are on Jan 01 2009.

This is a companion to the paper "The Toothpick Sequence" by David Applegate, Omar E. Pol and N. J. A. Sloane.

## The Sequences

Abbreviations: CA = cellular automaton, g.f. = generating function, nbrs = neighbors

Toothpick sequence, original version: A139250, A139251
Toothpick sequence, original version: limiting form of rows: A147646, A151688, A152980
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
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

Adamson's generalized toothpick family: A162958, A163267, A163311, A163312

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

Leftist toothpicks: A151566, A151565
Pascal triangle mod 2 (or Sierpinski triangle): A047999

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

X-toothpick sequence: A160170, A160171

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
Holladay-Ulam CA based on Maltese crosses: A151904, A151905, A151906, A151907

Z^2 with 4 nbrs: A147562, A147582
Z^2 with 4 nbrs, alternating states: A072272, A170878
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
Triangle-based CA: A161644, A161645, A170882, A170883

Z^3 with 6 nbrs: A151779, A151781
Z^3 with 26 nbrs: A160119, A160379

3-D toothpick structures: A160160, A160161, A160428, A170875, A170876, A170884, A170885

FCC lattice: A151776, A151777

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
(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)):
(a,b,c):
(1,1,0): A160573
(1,1,1): A151552
(1,1,2): A151692
(2,1,0): A151685
(2,1,1): A151691
(1,2,0): A151688 and A152980
(1,2,1): A151550
(2,2,0): A151693
(2,2,1): A151694
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
(a,b):
(1,2): A000012 and A000027
(1,3): A039966 and A005836
(1,4): A151666 and A000695
(1,5): A151667 and A033042
(2,2): A001316
(2,3): A151668
(2,4): A151669
(2,5): A151670
(3,2): A048883
(3,3): A117940
(3,4): A151665
(3,5): A151671
(4,2): A102376
(4,3): A151672
(4,4): A151673
(4,5): A151674

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

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