A364882 -id:A364882 - cp's OEIS frontend

A364392 a(1)=1 and thereafter a(n) is the least number of locations 1..n-1 which can be visited in a single path beginning at i=n-1, in which one proceeds from location i to i +- a(i) (within 1..n-1) until no further unvisited location is available.

1, 1, 2, 3, 4, 4, 3, 6, 3, 4, 4, 6, 3, 5, 4, 7, 5, 5, 6, 6, 5, 6, 6, 6, 6, 7, 3, 8, 5, 8, 7, 5, 6, 6, 7, 7, 9, 5, 9, 7, 5, 8, 7, 8, 3, 6, 9, 9, 7, 6, 4, 6, 6, 6, 10, 7, 7, 5, 10, 3, 6, 7, 7, 8, 3, 8, 6, 5, 9, 6, 4, 9, 9, 5, 7, 6, 5, 5, 7, 5, 6, 6, 6, 7, 7, 9, 7

Offset: 1

Neal Gersh Tolunsky, Jul 21 2023

The sequence is 1244 initial terms followed by a repeating block of 4925 terms so that a(n) = a(n-4925) for n >= 6170. - Kevin Ryde, Jul 31 2023

			a(13)=3 because beginning at the most recent location i=n-1=12, where a(12)=6, we can visit (the fewest possible) 3 locations in a single path as follows:
  1 2 3 4 5 6 7 8 9 10 11 12   location number i
  1,1,2,3,4,4,3,6,3, 4, 4, 6   a(i)
            <--------------6
            4-------->
At i=10, the only jump is back to 10-a(10) = 6, which was already visited, so the path stops.

Kevin Ryde, Table of n, a(n) for n = 1..10000
Kevin Ryde, PARI/GP code showing periodic
Index entries for linear recurrences with constant coefficients, order 4925.
Neal Gersh Tolunsky, Scatterplot of the ordinal transform of the first 6169 terms

Cf. A360593, A360746, A360745, A364882.

PARI
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More terms from Bert Dobbelaere, Jul 23 2023

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).

Original entry on oeis.org

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

Neal Gersh Tolunsky, Sep 09 2023

nonn

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.

			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}.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Cf. A364882, A364392, A360744, A362248.

cp's OEIS Frontend

A364392 a(1)=1 and thereafter a(n) is the least number of locations 1..n-1 which can be visited in a single path beginning at i=n-1, in which one proceeds from location i to i +- a(i) (within 1..n-1) until no further unvisited location is available.

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