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 A335973 The Locomotive Pushing or Pulling its Wagons sequence (see comments for definition). 4
 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 (list; graph; refs; listen; history; text; internal format)
 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: A119590 A266912 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

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Last modified October 23 17:37 EDT 2021. Contains 348215 sequences. (Running on oeis4.)