A335971 -id:A335971 - cp's OEIS frontend

A335972 The Locomotive Pushing the Wagons to the Right sequence (see Comments lines for definition).

2, 13, 14, 15, 16, 17, 18, 91, 201, 202, 31, 203, 24, 25, 26, 27, 28, 92, 41, 204, 35, 36, 37, 38, 93, 42, 51, 205, 46, 47, 48, 94, 52, 61, 206, 57, 58, 95, 62, 71, 207, 68, 96, 72, 81, 208, 97, 82, 131, 301, 302, 132, 133, 141, 303, 142, 134, 53, 143, 144, 63, 145, 64, 73, 146, 74

Offset: 1

Eric Angelini and Carole Dubois, Jul 03 2020

a(1) is the (pushing) locomotive; a(2), a(3), a(4),... a(n),... are the wagons. To hook a wagon to its successor (on the right) you must be able to insert the rightmost digit of a(n) between the first two digits of a(n+1). In mathematical terms, 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

			The sequence starts with 2, 13, 14, 15, 16, 17, 18, 91, 201, 202,...
a(1) = 2 as there is no earliest possible (pushing to the right) locomotive;
a(2) = 13 starts with 1 and 3: now 1 < 2 < 3 [2 being the rightmost digit of a(1)];
a(3) = 14 starts with 1 and 4: now 1 < 3 < 4 [3 being the rightmost digit of a(2)];
a(4) = 15 starts with 1 and 5: now 1 < 4 < 5 [4 being the rightmost digit of a(3)];
(...)
a(8) = 91 starts with 9 and 1: now 9 > 8 > 1 [8 being the rightmost digit of a(7)];
a(9) = 201 starts with 2 and 0: now 2 > 1 > 0 [1 being the rightmost digit of a(8)];
a(10) = 202 starts with 2 and 0: now 2 > 1 > 0 [1 being the rightmost digit of a(9)]; etc.

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

Cf. A335971 (locomotive pulling to the left) and A335973 (two locomotives).

def dead_end(k): return k%10 in {0, 9}
def aupto(n, seed=2):
  train, used = [seed], {seed}
  for n in range(2, n+1):
    caboose = train[-1]
    hook = caboose % 10
    low2, high2 = 10 + (hook + 1), 90 + (hook - 1)
    an, pow10b = low2, 1
    while an in used or dead_end(an): an += 1
    first2 = an//pow10b
    a2, b2 = divmod(first2, 10)
    while True:
      if a2 < hook < b2 or a2 > hook > b2:
        train.append(an)
        used.add(an)
        break
      if first2 > high2:
        pow10b *= 10
        an = low2*pow10b
      else:
        an += pow10b
        an -= an%pow10b
      while an in used or dead_end(an): an += 1
      first2 = an//pow10b
      a2, b2 = divmod(first2, 10)
  return train    # use train[n-1] for a(n)
print(aupto(66))  # Michael S. Branicky, Dec 14 2020

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

Original entry on oeis.org

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

Eric Angelini and Carole Dubois, Jul 03 2020

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

			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.

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

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

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

cp's OEIS Frontend

A335972 The Locomotive Pushing the Wagons to the Right sequence (see Comments lines for definition).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python