A360595 a(n) is the maximum number of locations 1..n-1 which can be visited in a single path starting from i = n-1, where jumps from location i to i +- a(i) are permitted (within 1..n-1) and a term can be visited up to three times.

0, 3, 1, 2, 2, 12, 1, 2, 2, 4, 2, 10, 15, 1, 2, 2, 4, 2, 10, 20, 1, 2, 2, 4, 2, 10, 13, 8, 2, 10, 2, 15, 7, 15, 25, 17, 53, 1, 2, 2, 4, 2, 10, 65, 1, 2, 2, 4, 2, 10, 13, 8, 2, 10, 2, 15, 7, 15, 72, 1, 2, 2, 4, 2, 10, 24, 18, 52

Offset

1,2

Comments

When a location is visited more than once, each such visit counts in a(n).

a(0)=0 is no terms before n=0 so an empty path.

Links

Table of n, a(n) for n=1..68.

Example

For n=6, the following is the longest chain of jumps starting from i = n-1 = 5,
  1  2  3  4  5   location number i
  0, 3, 1, 2, 2   a(i)
        1<----
         ->2
     3<----
      ------->2
        1<----
         ->2
     3<----
      ------->2
        1<----
         ->2
     3<----
It visited the terms 2,1,2,3 three times in a loop, which gives a total of 12 terms, so a(6)=12.

Prog

(Python)
def A(lastn, times=3, mode=0):
  a, n=[0], 0
  while n<lastn:
    d, i, v, o, g, r=[[n]], 0, 1, [], 0, 0
    while len(d)>0:
      if len(d[-1])>v: v, o=len(d[-1]), d[-1][:]
      if d[-1][-1]-a[d[-1][-1]]>=0:
        if d[-1].count(d[-1][-1]-a[d[-1][-1]])<times:g=1
      if d[-1][-1]+a[d[-1][-1]]<=n:
        if d[-1].count(d[-1][-1]+a[d[-1][-1]])<times:
          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
    a.append(v)
    n+=1
    if mode==0: print(n+1, a[n])
    if mode>0:
      u, q=0, []
      while u<len(o):
        q.append(a[o[u]])
        u+=1
      print(n+1, a[n], q, o)
  return a

Crossrefs

Cf. A360593.

Sequence in context: A201282 A234522 A057056 * A016469 A274342 A138746

Adjacent sequences: A360592 A360593 A360594 * A360596 A360597 A360598

Keyword

nonn

Author

S. Brunner, Feb 14 2023

Status

approved