A319610 a(n) is the minimal number of successive OFF cells that appears in n-th generation of rule-30 1D cellular automaton started from a single ON cell.
0, 0, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 0
Keywords
Examples
The Rule-30 1D cellular automaton started from a single ON (.) cell generates the following triangle: 1 . a(1)= (0) 2 . . . a(2)= (0) 3 . . 0 0 . a(3)= (2) 4 . . 0 . . . . a(4)= (1) 5 . . 0 0 . 0 0 0 . a(5)= (2) 6 . . 0 . . . . 0 . . . a(6)= (1) 7 . . 0 0 . 0 0 0 0 . 0 0 . a(7)= (2) 8 . . 0 . . . . 0 0 . . . . . . a(8)= (1) 9 . . 0 0 . 0 0 0 . . . 0 0 0 0 0 . a(9)= (2) 10 . . 0 . . . . 0 . . 0 0 . 0 0 0 . . . a(10)=(1) 11 . . 0 0 . 0 0 0 0 . 0 . . . . 0 . . 0 0 . a(11)=(1) 12 . . 0 . . . . 0 0 . . 0 . 0 0 0 0 . 0 . . . . a(12)=(1) 13 . . 0 0 . 0 0 0 . . . 0 0 . . 0 0 . . 0 . 0 0 0 . a(13)=(1)
Links
- Charlie Neder, Repeating pattern of length-1 runs
- Index entries for linear recurrences with constant coefficients, signature (1).
Crossrefs
Cf. A100053.
Programs
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Mathematica
CellularAutomaton[30, {{1}, 0}, 200]; (Reverse[Internal`DeleteTrailingZeros[Reverse[Internal`DeleteTrailingZeros[#]]]]) & /@ %; Table[Length /@ Select[%[[i]] // Split, Total[#] == 0 &] // Min, {i, 1, % // Length}]
Formula
G.f.: x (x + x/(1 - x) + x^3 + x^5 + x^7) (conjectured).
For n > 9, a(n)=1 at least up to n = 20000.
It is conjectured that for all n>=10, a(n)=1.
A period-4 pattern of length-1 runs starting at row 26 forces a(n) = 1 for all n >= 26 (see image). - Charlie Neder, Dec 15 2018
Comments