A362415 - OEIS

%I #40 Dec 19 2024 11:46:19

%S 1,2,1,1,2,2,1,1,1,2,1,1,1,1,2,2,2,1,1,1,1,1,2,2,2,2,1,1,1,1,1,1,2,2,

%T 2,2,2,1,1,1,1,1,1,1,2,2,2,2,2,2,1,1,1,2,2,2,2,1,1,1,1,1,1,1,1,2,2,2,

%U 2,2,2,2,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,2

%N The n-th run occurs a(n) times, where each run is chosen to be shortest, then lexicographically earliest.

%C When a run occurs 2 times, as run numbers r and s, it has both a(r)=2 and a(s)=2.

%C What are the densities of 1's and 2's?

%H Kevin Ryde, <a href="/A362415/b362415.txt">Table of n, a(n) for n = 1..10000</a>

%H Kevin Ryde, <a href="/A362415/a362415.gp.txt">PARI/GP Code</a>

%e The first few runs begin

%e      n: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

%e   a(n): 1, 2, 1, 1, 2, 2, 1, 1, 1, 2

%e    Run: 1, 2, 3,    4,    5,       6

%e Run r=1 is {1} and its a(r) = a(1) = 1 says this {1} occurs 1 time, so never again.

%e Run r=2 is {2} and its a(r) = a(2) = 2 says this {2} occurs 2 times, so again later.

%e Run r=3 cannot be {1} since a(1)=1 does not allow a second occurrence of {1}, but it can be {1,1} and its a(r) = a(3) = 1 says this {1,1} occurs 1 time, so never again.

%e Run r=4 cannot be {2} because although permitted by its previous a(2)=2, here its a(r) = a(4) = 1 would say {2} occurs 1 time (which would be false), so instead run 4 must be {2,2}.

%o (PARI) \\ See links.

%Y Cf. A361249 (run length transform), A000002, A001462, A028242.

%K nonn,changed

%O 1,2

%A _Neal Gersh Tolunsky_, Apr 19 2023