A266180 Decimal representation of the n-th iteration of the "Rule 6" elementary cellular automaton starting with a single ON (black) cell.

1, 6, 16, 96, 256, 1536, 4096, 24576, 65536, 393216, 1048576, 6291456, 16777216, 100663296, 268435456, 1610612736, 4294967296, 25769803776, 68719476736, 412316860416, 1099511627776, 6597069766656, 17592186044416, 105553116266496, 281474976710656

Offset

0,2

Comments

A001025 is a subsequence. - Altug Alkan, Dec 23 2015

Rules 38, 134 and 166 also generate this sequence.

References

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

Links

Robert Price, Table of n, a(n) for n = 0..999

Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Math around the Block (2022), p. 22.

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 (0,16).

Formula

From Colin Barker, Dec 23 2015 and Apr 13 2019: (Start)

a(n) = 4^(n-1)*(5-(-1)^n).

a(n) = 16*a(n-2) for n>1.

G.f.: (1+6*x) / ((1-4*x)*(1+4*x)).

(End)

Mathematica

rule=6; 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 *)
LinearRecurrence[{0, 16}, {1, 6}, 30] (* Harvey P. Dale, May 25 2016 *)

Prog

(Python) print([int(4**(n-1)*(5-(-1)**n)) for n in range(30)]) # Karl V. Keller, Jr., Jun 03 2021

Crossrefs

Cf. A001025, A266178, A266179, A019590, A003953, A003945, A000034, A032766, A042948, A035608.

Sequence in context: A056204 A091148 A210370 * A229566 A222965 A009354

Adjacent sequences: A266177 A266178 A266179 * A266181 A266182 A266183

Keyword

nonn,easy

Author

Robert Price, Dec 22 2015

Status

approved