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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).

%I A365576 #33 Sep 20 2023 10:00:03
%S A365576 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,
%T A365576 18,19,20,21,22,23,24,25,26,27,27,28,29,27,28,29,30,31,32,33,34,35,36,
%U A365576 37,38,39,40,41,42,43,44,45,46,47,48,49,50,50,51,52,53
%N 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).
%C A365576 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.
%C A365576 a(n) <= a(n-1) + 1.
%H A365576 Neal Gersh Tolunsky, <a href="/A365576/b365576.txt">Table of n, a(n) for n = 1..10000</a>
%e A365576 a(5)=3 because there are 3 distinct sets of locations which represent the indices reachable from a given location s.
%e A365576 Starting at s=1, we can visit the set of locations i = {1, 3}
%e A365576   1  2  3  4
%e A365576   2, 1, 2, 2
%e A365576   2---->2
%e A365576 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:
%e A365576   1  2  3  4
%e A365576   2, 1, 2, 2
%e A365576   2<----2
%e A365576 From s=2, we can visit the set of locations i = {1, 2, 3}:
%e A365576   1  2  3  4
%e A365576   2, 1, 2, 2
%e A365576   2<-1->2
%e A365576 From s=4, we can visit another distinct set of locations i = {1, 2, 3, 4}
%e A365576   1  2  3  4
%e A365576   2, 1, 2, 2
%e A365576      1<----2
%e A365576   2<-1->2
%e A365576 This gives a total of 3 distinct sets of locations reachable from any starting index (equivalent to 3 strongly connected components):
%e A365576   i = {1, 3}; i = {1, 2, 3}; and i = {1, 2, 3, 4}.
%Y A365576 Cf. A364882, A364392, A360744, A362248.
%K A365576 nonn
%O A365576 1,1
%A A365576 _Neal Gersh Tolunsky_, Sep 09 2023