A266218 Decimal representation of the n-th iteration of the "Rule 7" elementary cellular automaton starting with a single ON (black) cell.

1, 6, 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

Author

Robert Price, Dec 24 2015

Keywords

• nonn
• easy

References

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

Links

• Robert Price, Table of n, a(n) for n = 0..499
• 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).

Crossrefs

Cf. A266216, A266217.

Programs

Mathematica

rule=7; 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([(2*4**n-1)*(n%2) + 0**n - 0**abs(n-1) for n in range(33)]) # Karl V. Keller, Jr., Aug 17 2021

Formula

From Colin Barker, Dec 25 2015 and Apr 13 2019: (Start)

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

G.f.: (1+6*x-17*x^2+25*x^3+16*x^4-16*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)).

(End)

a(n) = (2*4^n - 1)*(n mod 2) + 0^n - 0^abs(n-1). - Karl V. Keller, Jr., Aug 17 2021