A266176 Decimal representation of the n-th iteration of the "Rule 5" elementary cellular automaton starting with a single ON (black) cell.

1, 2, 4, 107, 16, 1967, 64, 32447, 256, 523007, 1024, 8383487, 4096, 134197247, 16384, 2147401727, 65536, 34359410687, 262144, 549754503167, 1048576, 8796087779327, 4194304, 140737467383807, 16777216, 2251799729799167, 67108864, 36028796683419647, 268435456

Offset: 0

Author

Robert Price, Dec 22 2015

Keywords

• nonn
• easy

References

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

Links

• Robert Price, Table of n, a(n) for n = 0..999
• 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 (0,21,0,-84,0,64).

Crossrefs

Cf. A266174, A266175, A000079, A188530.

Programs

Mathematica

rule = 5; rows = 30; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]],2], {k,1,rows}]

Python

print([2*4**n -5*2**(n-1) -1 if n%2 else 2**n for n in range(30)]) # Karl V. Keller, Jr., Jun 20 2021

Formula

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

a(n) = 21*a(n-2) - 84*a(n-4) + 64*a(n-6) for n>5.

G.f.: (1+2*x-17*x^2+65*x^3+16*x^4-112*x^5) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-4*x)*(1+4*x)).

(End)

a(n) = 2^n = A000079(n) for even n>=0; a(n) = 2*4^n - 5*2^(n-1) - 1 = A188530(n) for odd n. - Karl V. Keller, Jr., Jun 19 2021