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A335971
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The Locomotive Pulling the Wagons to the Left sequence (see Comments lines for definition).
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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(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)
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CROSSREFS
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Cf. A335972 (locomotive pushing to the right) and A335973 (two locomotives).
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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