Successive triples, same pattern

Hello Math-Fun,

See the hereunder sequence S as a succession of triples (we use yellow and red colors to highlight them):

S = 0, 2, 1, 3, 5, 4, 6, 8, 7, 12, 13, 9, 10, 20, 14, 11, 21, 15, 16, 70, 17, 18, 90, 19, 22, 30, 23, 24, 50, 25, 26, 71, 27, 28, 91, 29, 31, 40, 34, 32, 41, 35, 33, 42, 36, 37, 80, 38, 43, 51, 39,...

0, 2, 1, 3, 5, 4, 6, 8, 7, 12, 13, 9, 10, 20, 14, 11, 21, 15, 16, 70, 17, 18, 90, 19, 22, 30, 23, 24, 50, 25, 26, 71, 27, 28, 91, 29, 31, 40, 34, 32, 41, 35, 33, 42, 36, 37, 80, 38, 43, 51, 39,...

There is an obvious pattern in those triples: the central integer n₂ in a triple is strictly bigger than its two neighbors.
We have  [n₁ < n₂ > n₃].

S is peculiar – as the same pattern is present in the successive triples of digits (we use below blue and cyan colors to highlight them): the central digit d₂ in a digit-triple is strictly bigger than its two neighbors.

We have [d₁ < d₂ > d₃]:

0, 2, 1, 3, 5, 4, 6, 8, 7, 12, 13, 9, 10, 20, 14, 11, 21, 15, 16, 70, 17, 18, 90, 19, 22, 30, 23, 24, 50, 25, 26, 71, 27, 28, 91, 29, 31, 40, 34, 32, 41, 35, 33, 42, 36, 37, 80, 38, 43, 51, 39,...

We wanted S to be the lexicographically earliest sequence of distinct nonnegative terms with this property.

Quiz: 
1) what is the smallest integer that will never be part of S?
2) what is the first wrong integer of S (as S was a nightmare to build by hand – my apologizes). 

Best,

É.

__________

P.-S.

More such patterns might be interesting to explore: [a > b < c], [a < b < c], [a > b > c], etc.

__________

Update, December 12

Maximilian was quick to answer both questions:

1) I think it's 99 because consecutive equal digits must be the last 

of one triple and the first of the next triple, but 9 can't be the

last nor the first of a triple because the central one must be larger.

Then the next "impossible" term is probably 109, where digits "10" 

must be the last two of a triple, but again, 9 can't be the first 

of the next triple.

110 is also impossible, again the two '11' must be last and first

of two consecutive triples, but 0 can't be the central one. 

In general, two equal digits must always be preceded and followed 

by a strictly larger digit. For that reason,  111 and 122 etc are 

also impossible.

2) > What is the first wrong integer of S (as S was a nightmare to 

   build by hand).

... There is no wrong integer in S! 

(Nor in your computation - at least according to my program. 

  Congratulations!)

[A special situation happens first after S(83)=78.

At first sight, the term 101 seems possible for S(84), but then the

next term would have to start with a digit 0, which is impossible.

So one has to take S(84) = 102, which is followed by 104, 103, 105, 

107, 106, 108, 109, 82, 83, 89, 84, 85, 120, 79, 86,...

after which we need a larger digit followed by a smaller digit.

87 and 93 ... 98 would be available, but for none of these there's 

a possible (smaller) successor (a number with two digits in increasing 

order) therefore the smallest possible successor is the very "early 

bird" 700.]

- Maximilian