%N Lexicographically earliest sequence of distinct nonnegative terms such that both a(n) and the cumulative sum a(1)+a(2)+...+a(n) have digits in nondecreasing order.
%C 10 is obviously the first integer not present in the sequence as 1 > 0.
%e Terms a(1) = 0 to a(5) = 5 sum up to 11: those six numbers have digits in nondecreasing order;
%e terms a(1) = 0 to a(6) = 4 sum up to 15: those seven numbers have digits in nondecreasing order;
%e terms a(1) = 0 to a(7) = 7 sum up to 22: those eight numbers have digits in nondecreasing order; etc.
%o def nondec(n): s = str(n); return s == "".join(sorted(s))
%o def aupton(terms):
%o alst = 
%o for n in range(2, terms+1):
%o an, cumsum = 1, sum(alst)
%o while True:
%o while an in alst: an += 1
%o if nondec(an) and nondec(cumsum + an): alst.append(an); break
%o else: an += 1
%o return alst
%o print(aupton(100)) # _Michael S. Branicky_, Mar 07 2021
%Y Cf. A009994 (numbers with digits in nondecreasing order), A342264 and A342266 (variations on the same idea).
%A _Eric Angelini_ and _Carole Dubois_, Mar 07 2021