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A365576
a(1)=2; thereafter a(n) is the number of strongly connected components in the digraph of the sequence thus far, where jumps from location i to i+-a(i) are permitted (within 1..n-1).
3
2, 1, 2, 2, 3, 2, 2, 3, 3, 4, 5, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 27, 28, 29, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53
OFFSET
1,1
COMMENTS
If two locations j and k can reach other, then they belong to the same strongly connected component and can reach the same set of locations.
a(n) <= a(n-1) + 1.
LINKS
EXAMPLE
a(5)=3 because there are 3 distinct sets of locations which represent the indices reachable from a given location s.
Starting at s=1, we can visit the set of locations i = {1, 3}
1 2 3 4
2, 1, 2, 2
2---->2
This is the same set of locations that can be visited from s=3. Since it is the same set, we only count it once:
1 2 3 4
2, 1, 2, 2
2<----2
From s=2, we can visit the set of locations i = {1, 2, 3}:
1 2 3 4
2, 1, 2, 2
2<-1->2
From s=4, we can visit another distinct set of locations i = {1, 2, 3, 4}
1 2 3 4
2, 1, 2, 2
1<----2
2<-1->2
This gives a total of 3 distinct sets of locations reachable from any starting index (equivalent to 3 strongly connected components):
i = {1, 3}; i = {1, 2, 3}; and i = {1, 2, 3, 4}.
CROSSREFS
KEYWORD
nonn
AUTHOR
Neal Gersh Tolunsky, Sep 09 2023
STATUS
approved