OCA:Rule 150

Rule 150
x=0, y = 0, rule = W150 ! #C [[ THEME Inverse ]] #C [[ RANDOMIZE2 THUMBLAUNCH THUMBNAIL THUMBSIZE 2 GRID ZOOM 6 WIDTH 600 HEIGHT 600 LABEL 90 -20 2 "#G" AUTOSTART PAUSE 2 GPS 8 LOOP 256 ]] LifeViewer-generated pseudorandom soup
Number	150
Computation	p⊕q⊕r
Character	Class 4
Black/white reversal	150
Left/right reflection	150
Strobing dual	105
Checkerboard dual	150

Rule 150 (or W150) is a 1-dimensional range-1 cellular automaton and parity rule.

Cells become the XOR of their neighbourhood. As such, its rule integer in binary (when read little-endian) is a subsection of the Thue-Morse sequence .

Current pattern	111	110	101	100	011	010	001	000
New state for center cell	1	0	0	1	0	1	1	0

Its equivalence class (together with its strobing dual) is one of three that are both symmetrical and self-complementary, together with those represented by 204 (the identity) and 232 (in which cells become the state of the majority of their neighbours (the 1D equivalent to Vote). It is one of two classes invariant under (not permuted by) the checkerboard dual (the other class being 204).

symmetrical self-complementary rules
s		s		s		s
	c			c	c	c	c
c
232	23	77	178	204	51	150	105

As a parity rule, it is distributive with XOR (and therefore unable to perform computations). Beginning with a single cell on the 0th column at t=0, the state of the cell in the kth column on the tth iteration is the coefficient of x^k (mod 2) in the expansion of (x^-1+1+x)^t.

When a single cell's evolution is read as a triangle, it forms sequence A071036. When concatenated to binary numbers, rows form A118110 (when converted to decimal numbers, A038184). There are A071053(t) cells on in the tth iteration, and A134659(t) on in all generations up to and including the tth.

In particular, since the sequence a(n) = A134659(2ⁿ-1) satisfies a(n) = 2*a(n-1)+4*a(n-2), a(n) = ∑2ⁿ-1t=0(A071053(t)) = F(n+2)*2ⁿ, where F(n) is the nth Fibonacci number.

The maximum period of a pattern in a cyclic rule 150 universe is A085588.

Oscillators of any period can be constructed in Rule 150 on finite-length lines with cylindrical, always-off, or always-on boundary conditions. In analogy to rule 110 unit cells being used to prove rules' universality, finding unit cells emulating rule 150 at one iteration/generation (occupying one column or two half-diagonals each) can prove rules' omniperiodicity (used by yujh for oscillators in B34kz5e7c8/S23-a4ityz5k and Matthias Merzenich for photons in B25678/S35678).

Rule 150's single-cell timespace diagram is formed by the 'shadow' of the lightcone (light-hexagonal-pyramid) in the parity rule in a triangular von Neumann neighbourhood, collapsed along its longest diameter.

See also

• Its 2D analogue, Fredkin (in which cells are XORed with 3 × 3 square neighbourhoods, and a single cell's evolution is a Cartesian product of that in rule 150 with itself), and other parity rule Replicator (in which cells exclude themselves from their neighbourhood)
• Rule 120, the only other rule that forms self-similar fractals which are not Sierpinski triangles

Further reading

• Jens Christian Claussen (2004). Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration
• Steven Finch, Pascal Sebah, Zai-Qiao Bai (2008). Odd Entries in Pascal's Trinomial Triangle
• N. J. A. Sloane (2015). On the Number of ON Cells in Cellular Automata
• Alan J. Macfarlane (2016). On generating functions of some sequences of integers defined in the evolution of the cellular automaton Rule 150

this was a follow-up to Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers