A335971 The Locomotive Pulling the Wagons to the Left sequence (see Comments lines for definition).

13, 20, 14, 24, 30, 15, 25, 31, 26, 35, 40, 16, 27, 36, 41, 28, 37, 42, 38, 46, 50, 17, 29, 39, 47, 51, 48, 52, 49, 53, 402, 18, 57, 60, 19, 58, 61, 59, 62, 302, 102, 103, 104, 105, 106, 107, 63, 403, 108, 64, 502, 109, 68, 70, 69, 71, 202, 113, 203, 114, 204, 115, 205, 116, 206

Offset: 1

Author

Eric Angelini and Carole Dubois, Jul 03 2020

Keywords

• base
• nonn

Comments

a(1) is the locomotive; a(2), a(3), a(4),... a(n),... are the successive wagons. To hook a wagon to its predecessor (on the left) you must be able to insert the leftmost digit of a(n) between the last 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 penultimate and the last digit of a(n-1). This is the lexicographically earliest sequence of distinct positive terms with this property.

Examples

			The sequence starts with 13, 20, 14, 24, 30,...
a(1) = 13 as there is no earliest possible (pulling to the left) locomotive;
a(2) = 20 starts with 2 and 1 < 2 < 3 [1 and 3 being the last two digits of a(1)];
a(3) = 14 starts with 1 and 2 > 1 > 0 [2 and 0 being the last two digits of a(2)];
a(4) = 24 starts with 2 and 1 < 2 < 4 [1 and 4 being the last two digits of a(3)]; etc.
		

Links

• Carole Dubois, Table of n, a(n) for n = 1..5001

Crossrefs

Cf. A335972 (locomotive pushing to the right) and A335973 (two locomotives).

Programs

Python

def dead_end(k): return abs((k//10)%10 - k%10) <= 1
def aupto(n, seed=13):
  train, used = [seed], {seed}
  for n in range(2, n+1):
    caboose = train[-1]
    h1, h2 = sorted([(caboose//10)%10, caboose%10])
    hooks = set(range(h1+1, h2))
    pow10 = 10
    an = min(hooks)*pow10
    while an in used: an += 1
    hook = an//pow10
    while True:
      if hook in hooks:
        if not dead_end(an):
          train.append(an)
          used.add(an)
          break
      else: pow10 *= 10
      an = max(an+1, min(hooks)*pow10)
      while an in used: an += 1
      hook = an//pow10
  return train    # use train[n-1] for a(n)
print(aupto(65))  # Michael S. Branicky, Dec 14 2020