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A265721
Decimal representation of the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.
5
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
0,3
COMMENTS
Rule 33 also generates this sequence.
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
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
EXAMPLE
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)
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}]
PROG
(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
KEYWORD
nonn,easy
AUTHOR
Robert Price, Dec 14 2015
STATUS
approved