OFFSET
1,2
LINKS
Charlie Neder, Repeating pattern of length-1 runs
Index entries for linear recurrences with constant coefficients, signature (1).
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
EXAMPLE
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)
MATHEMATICA
CellularAutomaton[30, {{1}, 0}, 100];
(Reverse[Internal`DeleteTrailingZeros[
Reverse[Internal`DeleteTrailingZeros[#]]]]) & /@ %;
Table[Length /@ Select[%[[i]] // Split, Total[#] > 0 &] // Min, {i,
1, % // Length}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Philipp O. Tsvetkov, Sep 25 2018
STATUS
approved