cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A364882 a(1)=1 and thereafter a(n) is the number of locations 1..n-1 which are visited last in a single path beginning at some location s, in which one proceeds from location i to i +- a(i) (within 1..n-1) until no further unvisited location is available.

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 3, 4, 6, 6, 7, 7, 7, 7, 7, 7, 9, 9, 9, 11, 11, 11, 14, 15, 15, 15, 15, 17, 18, 18, 18, 19, 19, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27, 28, 28, 28, 28, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 30, 30, 30, 30, 30, 30, 40, 40
Offset: 1

Views

Author

Neal Gersh Tolunsky, Aug 11 2023

Keywords

Comments

A location can be visited no more than once in a single path.

Examples

			a(9)=6 because there are 6 locations which can be visited last (as a dead end) among i=1..8. The 6 locations are i=1,2,3,5,7,8. The following shows a path in which the last location is i=5, beginning at location s=8:
  1 2 3 4 5 6 7 8   location number i
  1,1,2,3,3,3,3,4   a(i)
  1<----3<------4
  1>1>2-->3
From i=5, the only jumps are back to i=1 or forward to i=8, both of which were already visited, so i=5 is one possible dead end term. Here is a path illustrating how i=7 can be a dead end term. We begin at s=4.
  1 2 3 4 5 6 7 8   location number i
  1,1,2,3,3,3,3,4   a(i)
        3---->3
From i=7, we can only jump back to i=4, which was already visited, so i=7 is a dead end term. There are 4 other locations which can be last (or dead ends), for a total of 6 such locations, so a(9)=6.

Links

Crossrefs

Cf. A364392, A360744, A360593.

Extensions

More terms from Kevin Ryde, Aug 26 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

Views

Author

Neal Gersh Tolunsky, Sep 09 2023

Keywords

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.

Examples

			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}.

Links

Crossrefs

Cf. A364882, A364392, A360744, A362248.

A367128 a(1)=a(2)=1; thereafter a(n) is the radius of the sequence's digraph, where jumps from location i to i+-a(i) are permitted (within 1..n-1).

Original entry on oeis.org

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

Views

Author

Neal Gersh Tolunsky, Nov 05 2023

Keywords

Comments

The radius of the sequence's digraph is the smallest eccentricity of any vertex (location) in the graph. The eccentricity of a location i means the largest number of jumps in the shortest path from location i to any other location.

Examples

			To find a(5), we can look at the eccentricity of each location:
  i            = 1     2     3     4
  a(i)         = 1,    1,    1,    1
                 1 <-> 1 <-> 1 <-> 1
  eccentricity = 3     2     2     3
i=1 has eccentricity 3 because it requires up to 3 jumps to reach any other location (3 to i=4), and similarly i=4 has eccentricity 3 too.
i=2 and i=3 have eccentricity 2 as they require at most 2 jumps to reach anywhere.
The smallest eccentricity of any location is 2, which makes 2 the radius of the sequence's digraph, so a(5)=2.

Links

Crossrefs

Cf. A367129, A365576, A364392, A362248, A360744, A360745, A360746.

Programs

  • C
    /* See links */

A367129 a(1)=a(2)=1; thereafter a(n) is the diameter of the sequence's digraph, where jumps from location i to i+-a(i) are permitted (within 1..n-1).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 4, 4, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 24, 24, 24, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22
Offset: 1

Views

Author

Neal Gersh Tolunsky, Nov 05 2023

Keywords

Comments

The diameter of the sequence's digraph is the largest eccentricity of any vertex (location) in the graph. The eccentricity of a location i means the largest number of jumps in the shortest path from location i to any other location.

Examples

			a(5)=3 because i=1 has the largest eccentricity of any location. i=1 takes 3 jumps to reach i=4 in the shortest path:
  i    = 1  2  3  4
  a(i) = 1, 1, 1, 2
         1->1->1->2
Every other location has eccentricity 2, which makes 3 the largest eccentricity and thus the diameter of the sequence's digraph, so a(5)=3.

Links

Crossrefs

Cf. A367128, A365576, A364392, A362248, A360744, A360745, A360746.

Programs

  • C
    /* See links */
Showing 1-4 of 4 results.

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