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

%I A335971 #19 Dec 14 2020 01:35:57
%S A335971 13,20,14,24,30,15,25,31,26,35,40,16,27,36,41,28,37,42,38,46,50,17,29,
%T A335971 39,47,51,48,52,49,53,402,18,57,60,19,58,61,59,62,302,102,103,104,105,
%U A335971 106,107,63,403,108,64,502,109,68,70,69,71,202,113,203,114,204,115,205,116,206
%N A335971 The Locomotive Pulling the Wagons to the Left sequence (see Comments lines for definition).
%C A335971 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.
%H A335971 Carole Dubois, <a href="/A335971/b335971.txt">Table of n, a(n) for n = 1..5001</a>
%e A335971 The sequence starts with 13, 20, 14, 24, 30,...
%e A335971 a(1) = 13 as there is no earliest possible (pulling to the left) locomotive;
%e A335971 a(2) = 20 starts with 2 and 1 < 2 < 3 [1 and 3 being the last two digits of a(1)];
%e A335971 a(3) = 14 starts with 1 and 2 > 1 > 0 [2 and 0 being the last two digits of a(2)];
%e A335971 a(4) = 24 starts with 2 and 1 < 2 < 4 [1 and 4 being the last two digits of a(3)]; etc.
%o A335971 (Python)
%o A335971 def dead_end(k): return abs((k//10)%10 - k%10) <= 1
%o A335971 def aupto(n, seed=13):
%o A335971   train, used = [seed], {seed}
%o A335971   for n in range(2, n+1):
%o A335971     caboose = train[-1]
%o A335971     h1, h2 = sorted([(caboose//10)%10, caboose%10])
%o A335971     hooks = set(range(h1+1, h2))
%o A335971     pow10 = 10
%o A335971     an = min(hooks)*pow10
%o A335971     while an in used: an += 1
%o A335971     hook = an//pow10
%o A335971     while True:
%o A335971       if hook in hooks:
%o A335971         if not dead_end(an):
%o A335971           train.append(an)
%o A335971           used.add(an)
%o A335971           break
%o A335971       else: pow10 *= 10
%o A335971       an = max(an+1, min(hooks)*pow10)
%o A335971       while an in used: an += 1
%o A335971       hook = an//pow10
%o A335971   return train    # use train[n-1] for a(n)
%o A335971 print(aupto(65))  # _Michael S. Branicky_, Dec 14 2020
%Y A335971 Cf. A335972 (locomotive pushing to the right) and A335973 (two locomotives).
%K A335971 base,nonn
%O A335971 1,1
%A A335971 _Eric Angelini_ and _Carole Dubois_, Jul 03 2020