A099423 -id:A099423 - cp's OEIS frontend

A102127 Interval between successive arrivals at the origin of an LQTL CA.

4, 4, 4, 16, 4, 4, 4, 16, 4, 4, 4, 4, 4, 40, 4, 16, 4, 4, 4, 4, 12, 4, 4, 8, 4, 40, 4, 4, 4, 4, 4, 88, 4, 40, 4, 4, 4, 4, 4, 16, 4, 4, 4, 16, 4, 4, 4, 4, 4, 40, 4, 88, 4, 4, 4, 4, 4, 40, 4, 8, 4, 12, 4, 4, 4, 4, 4, 16, 4, 40, 4, 4, 4, 4, 4, 16, 4, 4, 4, 16, 4

Offset: 0

Robert H Barbour, Feb 14 2005

nonn

The "LQTL CA" is similar to the Langton's ant but has 4 states. The ant turns right from the cells with states 0, 1 and left from the cells with states 2, 3 and changes the state of the cell according to the rules 0 -> 1 -> 2 -> 3 -> 0. - Andrey Zabolotskiy, Jan 06 2023

Andrey Zabolotskiy, Table of n, a(n) for n = 0..999
Robert H. Barbour, Two-dimensional Four Color Cellular Automaton: Surface Explorations, Complex Systems, 17 (2007), 103-112.
Palash Sarkar, A Brief History of Cellular Automata, ACM Computing Surveys, Vol. 32, No. 1, March 2000, 80-107.

Cf. A094266, A094867, A099423, A274369, A102369.

First differences of A102110.

x = y = direction = 0
cells, a = {}, []
for n in range(1, 1000):
    c = cells.get((x, y), 0)
    cells[(x, y)] = (c + 1) % 4
    direction += [1, 1, -1, -1][c]
    (dx, dy) = [(1, 0), (0, 1), (-1, 0), (0, -1)][direction%4]
    x += dx;  y += dy
    if (x, y) == (0, 0):
        a.append(n)
print(a) # A102110
print([x-y for x, y in zip(a, [0]+a)]) # this sequence -  Andrey Zabolotskiy, Jan 06 2023

Terms a(26) and beyond from Andrey Zabolotskiy, Jan 06 2023

A094266 LQTL Lean Quaternary Temporal Logic: a terse form of temporal logic created by assigning four descriptors such that false, becoming true, true and becoming false are represented and become a linear sequence. In a branching tree two alternative are open, change or no change. The integer sequence above is the count of the row possibilities of the four states over successive iterations.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 1, 0, 1, 3, 3, 1, 2, 4, 6, 4, 6, 6, 10, 10, 16, 12, 16, 20, 36, 28, 28, 36, 72, 64, 56, 64, 136, 136, 120, 120, 256, 272, 256, 240, 496, 528, 528, 496, 992, 1024, 1056, 1024, 2016, 2016, 2080, 2080, 4096, 4032, 4096, 4160, 8256, 8128, 8128, 8256, 16512

Offset: 0

Robert H Barbour and L. D. Painter, Jun 01 2004

This is a table read by rows of length 4. Every row is formed from the previous one by the circular Pascal triangle-like rule: a, b, c, d -> d+a, a+b, b+c, c+d. Consider a labeled binary tree such that the root has label 0 and every node labeled k has children labeled k and (k+1) mod 4; the n-th row of this sequence counts nodes on the level n+1 with labels 0, 1, 2, 3, while the n-th row of A099423 counts nodes up to level n. - Andrey Zabolotskiy, Jan 06 2023

Robert H. Barbour, 2D Four-Color Cellular Automaton, NKS 2006 Wolfram Science Conference.
Robert H. Barbour, Two-dimensional Four Color Cellular Automaton: Surface Explorations, Complex Systems, 17 (2007), 103-112.

Cf. A000749, A005418, A038503, A038504, A038505, A020522, A063376.

Algorithm available from Robert H Barbour

Appears to satisfy a 12-degree linear recurrence. - Ralf Stephan, Dec 04 2004

A102110 Iterations during which LQTL cellular automaton passes through the origin.

Original entry on oeis.org

4, 8, 12, 28, 32, 36, 40, 56, 60, 64, 68, 72, 76, 116, 120, 136, 140, 144, 148, 152, 164, 168, 172, 180, 184, 224, 228, 232, 236, 240, 244, 332, 336, 376, 380, 384, 388, 392, 396, 412, 416, 420, 424, 440, 444, 448, 452, 456, 460, 500, 504, 592, 596, 600, 604

Offset: 1

Robert H Barbour, Feb 14 2005

Robert H. Barbour, Two-dimensional Four Color Cellular Automaton: Surface Explorations, Complex Systems, 17 (2007), 103-112.
Palash Sarkar, A Brief History of Cellular Automata, ACM Computing Surveys, Vol. 32 (2000), No. 1 March, p. 80-107.

Cf. A094266, A094867, A099423.

Cumulative sums of A102127.

Terms a(19) and beyond from Andrey Zabolotskiy, Jan 06 2023

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A102127 Interval between successive arrivals at the origin of an LQTL CA.

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A102110 Iterations during which LQTL cellular automaton passes through the origin.

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