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A362415
The n-th run occurs a(n) times, where each run is chosen to be shortest, then lexicographically earliest.
2
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
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
KEYWORD
nonn
AUTHOR
Neal Gersh Tolunsky, Apr 19 2023
STATUS
approved