A319658 a(n) is the minimal number of successive ON cells that appears in n-th generation of rule-30 1D cellular automaton started from a single ON cell.
1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 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
Offset: 1
Keywords
Examples
The Rule-30 1D cellular automaton started from a single ON (.) cell generates the following triangle: 1 . a(1)= (1) 2 . . . a(2)= (3) 3 . . 0 0 . a(3)= (1) 4 . . 0 . . . . a(4)= (2) 5 . . 0 0 . 0 0 0 . a(5)= (1) 6 . . 0 . . . . 0 . . . a(6)= (2) 7 . . 0 0 . 0 0 0 0 . 0 0 . a(7)= (1) 8 . . 0 . . . . 0 0 . . . . . . a(8)= (2) 9 . . 0 0 . 0 0 0 . . . 0 0 0 0 0 . a(9)= (1) 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).
Programs
-
Mathematica
CellularAutomaton[30, {{1}, 0}, 100]; (Reverse[Internal`DeleteTrailingZeros[ Reverse[Internal`DeleteTrailingZeros[#]]]]) & /@ %; Table[Length /@ Select[%[[i]] // Split, Total[#] > 0 &] // Min, {i, 1, % // Length}]
Formula
G.f.: 1/(1 - x) + 2 x + x^3 + x^5 + x^7 + x^13 (conjectured).
For n > 14, a(n)=1 at least until n = 10000.
It is conjectured that for all n >= 15, a(n)=1.
A period-4 pattern of length-1 runs beginning on row 19 forces a(n) = 1 for all n >= 19 (see image). - Charlie Neder, Dec 15 2018