A265721 Decimal representation of the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.
1, 0, 4, 99, 16, 1935, 64, 32319, 256, 522495, 1024, 8381439, 4096, 134189055, 16384, 2147368959, 65536, 34359279615, 262144, 549753978879, 1048576, 8796085682175, 4194304, 140737458995199, 16777216, 2251799696244735, 67108864, 36028796549201919, 268435456
Offset: 0
Examples
From _Michael De Vlieger_, Dec 14 2015: (Start)
First 8 rows, replacing leading zeros with ".", the row converted to its binary (A265720), then decimal equivalent at right:
1 -> 1 = 1
. . 0 -> 0 = 0
. . 1 0 0 -> 100 = 4
1 1 0 0 0 1 1 -> 1100011 = 99
. . . . 1 0 0 0 0 -> 10000 = 16
1 1 1 1 0 0 0 1 1 1 1 -> 11110001111 = 1935
. . . . . . 1 0 0 0 0 0 0 -> 1000000 = 64
1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 -> 111111000111111 = 32319
(End)
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
- 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
-
Mathematica
rule = 1; 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 - 7*2**(n-1) - 1 if n%2 else 2**n for n in range(50)]) # Karl V. Keller, Jr., Aug 24 2021
Formula
From Colin Barker, Dec 14 2015 and Apr 16 2019: (Start)
a(n) = 21*a(n-2) - 84*a(n-4) + 64*a(n-6) for n>5.
G.f.: (1-17*x^2+99*x^3+16*x^4-144*x^5) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-4*x)*(1+4*x)).
(End)
a(n) = 2*4^n - 7*2^(n-1) - 1 for odd n; a(n) = 2^n for even n. - Karl V. Keller, Jr., Aug 24 2021
Comments