A267685 Decimal representation of the n-th iteration of the "Rule 203" elementary cellular automaton starting with a single ON (black) cell.

1, 4, 27, 119, 495, 2015, 8127, 32639, 130815, 523775, 2096127, 8386559, 33550335, 134209535, 536854527, 2147450879, 8589869055, 34359607295, 137438691327, 549755289599, 2199022206975, 8796090925055, 35184367894527, 140737479966719, 562949936644095

Offset: 0

Author

Robert Price, Jan 19 2016

Keywords

• nonn
• easy

Comments

Conjectures from Barker confirmed by later formulas. - Ray Chandler, Aug 09 2025

References

• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

• Robert Price, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• S. Wolfram, A New Kind of Science
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata
• Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Crossrefs

Cf. A129868, A267683, A267684.

Programs

Mathematica

rule=203; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Python

print([1, 4]+[2*4**n - 2**n - 1 for n in range(2, 50)]) # Karl V. Keller, Jr., Jun 07 2022

Formula

From Colin Barker, Jan 19 2016 and Apr 17 2019: (Start)

a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>4.

G.f.: (1-3*x+13*x^2-22*x^3+8*x^4) / ((1-x)*(1-2*x)*(1-4*x)).

(End)

a(n) = A129868(n) for n >= 2. - Georg Fischer, Mar 26 2019

a(n) = 2*4^n - 2^n - 1 for n > 1. - Karl V. Keller, Jr., Jun 07 2022