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

Author

Robert Price, Dec 22 2015

Keywords

• nonn
• easy

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, Green Blue Mathematics (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).

Crossrefs

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

Programs

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 *)

Python

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

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)