A335973 The Locomotive Pushing or Pulling its Wagons sequence (see comments for definition).

13, 24, 35, 46, 57, 68, 791, 202, 14, 25, 36, 47, 58, 691, 203, 15, 26, 37, 48, 591, 204, 16, 27, 38, 491, 205, 17, 28, 391, 206, 18, 291, 207, 181, 208, 191, 302, 131, 2002, 135, 461, 303, 141, 304, 151, 305, 161, 306, 171, 307, 182, 31, 2003, 142, 308, 192, 41, 2004, 152, 313, 241, 2005, 162, 51, 402

Offset

1,1

Comments

a(1) is the locomotive; a(2), a(3), a(4),... a(n),... are the wagons. To hook a wagon both to its predecessor (on the left) and successor (on the right) you must be able to insert the leftmost digit of a(n) between the last two digits of a(n-1) AND to insert the rightmost digit of a(n) between the first two digits of a(n+1). In mathematical terms, the value of the leftmost digit of a(n) must be between (not equal to) the last two digits of a(n-1) AND the value of the rightmost digit of a(n) must be between (not equal to) the first and the second digit of a(n+1). This is the lexicographically earliest sequence of distinct positive terms with this property.

a(n) cannot end in 0 or 9. - Michael S. Branicky, Dec 14 2020

Links

Carole Dubois, Table of n, a(n) for n = 1..5006

Example

The sequence starts with 13, 24, 35, 46, 57, 68, 791, 202, 14,...
a(1) = 13 as there is no earliest possible locomotive;
a(2) = 24 starts with 2 and 4: now 2 < 3 < 4 [3 being the rightmost digit of a(1)] AND 3 < 4 < 5 [4 being the rightmost digit of a(2), 3 and 5 being the first two digits of a(3);
a(3) = 35 starts with 3 and 5: now 3 < 4 < 5 [4 being the rightmost digit of a(2)] AND 4 < 5 < 6 [5 being the rightmost digit of a(3), 4 and 6 being the first two digits of a(4);
a(4) = 46 starts with 4 and 6: now 4 < 5 < 6 [5 being the rightmost digit of a(3)] AND 5 < 6 < 7 [6 being the rightmost digit of a(4), 5 and 7 being the first two digits of a(5);
a(5) = 57 starts with 5 and 7: now 5 < 6 < 7 [6 being the rightmost digit of a(4)] AND 6 < 7 < 8 [7 being the rightmost digit of a(5), 6 and 8 being the first two digits of a(6);
a(6) = 68 starts with 6 and 8: now 6 < 7 < 8 [7 being the rightmost digit of a(5)] AND 7 < 8 < 9 [8 being the rightmost digit of a(6), 7 and 9 being the first two digits of a(7);
a(7) = 791 starts with 7 and 9: now 7 < 8 < 9 [8 being the rightmost digit of a(6)] AND 2 > 1 > 0 [1 being the rightmost digit of a(7); 2 and 0 being the first two digits of a(8); etc.

Prog

(Python)
def between(i, j, k):
  return i < j < k or i > j > k
def dead_end(k):
  rest, last = divmod(k, 10)
  if last in {0, 9}: return True
  return abs(rest%10 - last) <= 1
def aupto(n, seed=13):
  train, used = [seed], {seed}
  for n in range(2, n+1):
    caboose = train[-1]
    cabbody, cabhook = divmod(caboose, 10)
    h1, h2 = sorted([cabbody%10, cabhook])
    hooks = set(range(h1+1, h2))
    pow10 = 10
    an = min(hooks)*pow10
    while an in used or dead_end(an): an += 1
    hook = an//pow10
    while True:
      if hook in hooks:
        if between(hook, cabhook, an//(pow10//10)%10):
          train.append(an)
          used.add(an)
          break
      else: pow10 *= 10
      an = max(an+1, min(hooks)*pow10)
      while an in used or dead_end(an): an += 1
      hook = an//pow10
  return train    # use train[n-1] for a(n)
print(aupto(65))  # Michael S. Branicky, Dec 14 2020

Crossrefs

Cf. A335971 (locomotive pulling to the left) and A335972 (locomotive pushing to the right).

Sequence in context: A266912 A363823 A032607 * A190040 A081723 A189325

Adjacent sequences: A335970 A335971 A335972 * A335974 A335975 A335976

Keyword

base,nonn

Author

Eric Angelini and Carole Dubois, Jul 03 2020

Status

approved