A362415 The n-th run occurs a(n) times, where each run is chosen to be shortest, then lexicographically earliest.

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, 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, 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

Offset

1,2

Comments

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

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

Links

Kevin Ryde, Table of n, a(n) for n = 1..10000

Kevin Ryde, PARI/GP Code

Example

The first few runs begin
     n: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
  a(n): 1, 2, 1, 1, 2, 2, 1, 1, 1, 2
   Run: 1, 2, 3,    4,    5,       6
Run r=1 is {1} and its a(r) = a(1) = 1 says this {1} occurs 1 time, so never again.
Run r=2 is {2} and its a(r) = a(2) = 2 says this {2} occurs 2 times, so again later.
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.
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}.

Prog

(PARI) \\ See links.

Crossrefs

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

Sequence in context: A306615 A037826 A079882 * A317335 A014709 A278161

Adjacent sequences: A362412 A362413 A362414 * A362416 A362417 A362418

Keyword

nonn,changed

Author

Neal Gersh Tolunsky, Apr 19 2023

Status

approved