A266379 Binary representation of the n-th iteration of the "Rule 21" elementary cellular automaton starting with a single ON (black) cell.

1, 11, 0, 1111111, 0, 11111111111, 0, 111111111111111, 0, 1111111111111111111, 0, 11111111111111111111111, 0, 111111111111111111111111111, 0, 1111111111111111111111111111111, 0, 11111111111111111111111111111111111, 0, 111111111111111111111111111111111111111

Offset

0,2

Links

Robert Price, Table of n, a(n) for n = 0..500

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

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

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Index entries for linear recurrences with constant coefficients, signature (0,10001,0,-10000).

Formula

From Colin Barker, Dec 29 2015: (Start)

a(n) = 10001*a(n-2) - 10000*a(n-4) for n>5.

G.f.: (1 + 11*x - 10001*x^2 + 1001100*x^3 + 10000*x^4 - 1000000*x^5) / ((1-x)*(1+x)*(1-100*x)*(1+100*x)). (End)

a(n) = (1-(-1)^n)*(1000*10000^floor(n/2)-1)/18 for n>1. - Bruno Berselli, Dec 29 2015

a(n) = (10*100^n - 1)/9*(n mod 2) + 0^n - 100*0^abs(n-1). - Karl V. Keller, Jr., Sep 03 2021

Mathematica

rule=21; 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]]], {k, 1, rows}] (* Binary Representation of Rows *)

Prog

(Magma) [n le 1 select 11^n else (1-(-1)^n)*(1000*10000^Floor(n/2)-1)/18: n in [0..40]]; // Bruno Berselli, Dec 29 2015
(Python) print([(10*100**n - 1)//9*(n%2) + 0**n - 100*0**abs(n-1) for n in range(50)]) # Karl V. Keller, Jr., Sep 03 2021

Crossrefs

Cf. A266377, A266380.

Sequence in context: A036931 A165399 A157712 * A158215 A294254 A216792

Adjacent sequences: A266376 A266377 A266378 * A266380 A266381 A266382

Keyword

nonn,easy

Author

Robert Price, Dec 28 2015

Status

approved