A360744 a(n) is the maximum number of locations 1..n-1 which can be reached starting from some location s, where jumps from location i to i +- a(i) are permitted (within 1..n-1); a(1)=1. See example.

1, 1, 2, 3, 4, 5, 5, 6, 6, 7, 7, 9, 10, 10, 10, 11, 11, 13, 14, 14, 14, 15, 15, 15, 15, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 24, 24, 26, 27, 28, 29, 29, 29, 29, 29, 29, 29, 29, 29, 32, 32, 32, 32, 33, 33, 35, 35, 41, 42, 42, 42, 43, 43, 43, 44, 44, 45, 45, 46, 46, 46, 46, 46, 46, 47, 47, 49, 49, 51, 51, 51

Offset: 1

Author

Neal Gersh Tolunsky, Feb 18 2023

Keywords

• nonn

Comments

a(10)=7 is the earliest term whose solution cannot be represented by a single path in which each index is visited once.

Examples

			For a(9), we reach the greatest number of terms by starting at location s=4, which is a(4)=3. We visit 6 terms as follows (each line shows the next unvisited term(s) we can reach from the term(s) last visited):
1, 1, 2, 3, 4, 5, 5, 6
1<-------3------->5
1, 1, 2, 3, 4, 5, 5, 6
1->1<-------------5
1, 1, 2, 3, 4, 5, 5, 6
   1->2
1, 1, 2, 3, 4, 5, 5, 6
      2---->4
From the last iteration we can visit no new terms. We reached 6 terms, so a(9)=6:
1, 1, 2, 3, 4, 5, 5, 6
1  1  2  3  4     5
		

Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A360745, A360746, A360593, A361383, A359005, A358838, A359008, A362248.

Programs

Python

def A(lastn,mode=0):
  a,n,t=[1],0,1
  while n0:
        if not d[-1][-1] in rr:rr.append(d[-1][-1])
        if d[-1][-1]-a[d[-1][-1]]>=0:
          if d[-1].count(d[-1][-1]-a[d[-1][-1]])0: d.append(d[-1][:])
            d[-1].append(d[-1][-1]+a[d[-1][-1]])
            r=1
        if g>0:
          if r>0: d[-2].append(d[-2][-1]-a[d[-2][-1]])
          else: d[-1].append(d[-1][-1]-a[d[-1][-1]])
          r=1
        if r==0:d.pop()
        r,g=0,0
      if v0: print(a)
  return a ## S. Brunner, Feb 19 2023