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
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).
Programs
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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