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

1,3

Comments

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

Links

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

Example

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

Prog

(Python)
def A(lastn, mode=0):
  a, n, t=[1], 0, 1
  while n<lastn:
    p, v=0, 1
    while p<=n:
      d, g, r, rr=[[p]], 0, 0, [p]
      while len(d)>0:
        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]])<t:g=1
        if d[-1][-1]+a[d[-1][-1]]<=n:
          if d[-1].count(d[-1][-1]+a[d[-1][-1]])<t:
            if g>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 v<len(rr):v=len(rr)
      p+=1
    a.append(v)
    n+=1
    print(n+1, a[n])
    if mode>0: print(a)
  return a ## S. Brunner, Feb 19 2023

Crossrefs

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

Sequence in context: A365339 A046108 A079411 * A198454 A063882 A097873

Adjacent sequences: A360741 A360742 A360743 * A360745 A360746 A360747

Keyword

nonn

Author

Neal Gersh Tolunsky, Feb 18 2023

Status

approved