A266380 Decimal representation of the n-th iteration of the "Rule 21" elementary cellular automaton starting with a single ON (black) cell.

1, 3, 0, 127, 0, 2047, 0, 32767, 0, 524287, 0, 8388607, 0, 134217727, 0, 2147483647, 0, 34359738367, 0, 549755813887, 0, 8796093022207, 0, 140737488355327, 0, 2251799813685247, 0, 36028797018963967, 0, 576460752303423487, 0, 9223372036854775807, 0

Offset

0,2

References

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

Links

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

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Index entries for linear recurrences with constant coefficients, signature (0,17,0,-16).

Formula

From Colin Barker, Dec 29 2015 and Apr 15 2019: (Start)

a(n) = 17*a(n-2) - 16*a(n-4) for n>5.

G.f.: (1 + 3*x - 17*x^2 + 76*x^3 + 16*x^4 - 64*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)). (End)

a(n) = (1 - (-1)^n)*(4*16^floor(n/2) - 1/2) for n>1. - Bruno Berselli, Dec 29 2015

a(n) = (2*4^n - 1)*(n mod 2) + 0^n - 4*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]], 2], {k, 1, rows}] (* Decimal Representation of Rows *)

Prog

(Magma) [n le 1 select 3^n else (1-(-1)^n)*(4*16^Floor(n/2)-1/2): n in [0..40]]; // Bruno Berselli, Dec 29 2015
(Python) print([(2*4**n - 1)*(n%2) + 0**n - 4*0**abs(n-1) for n in range(50)]) # Karl V. Keller, Jr., Sep 03 2021

Crossrefs

Cf. A266377, A266379.

Cf. A241955: a(2*n+1) for n>0.

Sequence in context: A322458 A370712 A372314 * A076951 A359561 A060282

Adjacent sequences: A266377 A266378 A266379 * A266381 A266382 A266383

Keyword

nonn,easy

Author

Robert Price, Dec 28 2015

Status

approved