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A302503 Lexicographically first sequence of distinct terms such that any set of eight successive digits can be reordered as {d, d+1, d+2, d+3, d+4, d+5, d+6, d+7}, d being the smallest of the eight digits.

%I A302503 #18 Jul 23 2025 15:59:18
%S A302503 0,1,2,3,4,5,6,7,8,9,23,45,67,81,234,56,70,12,34,567,89,2345,678,92,
%T A302503 345,6781,23456,78,123,456,701,234567,812,3456,781,2345670,1234,5670,
%U A302503 12345,670,123456,789,2345678,923,4567,892,34567,8123,45670,1234567,8923,45678,9234,5678,92345,6789,23456781,23456701
%N A302503 Lexicographically first sequence of distinct terms such that any set of eight successive digits can be reordered as {d, d+1, d+2, d+3, d+4, d+5, d+6, d+7}, d being the smallest of the eight digits.
%C A302503 As the digit 0 has no predecessor and the digit 9 has no successor here, sets of successive digits like {6,5,4,3,2,1,0,9} and {3,4,5,6,7,8,9,0} are forbidden.
%H A302503 Dominic McCarty, <a href="/A302503/b302503.txt">Table of n, a(n) for n = 1..10000</a>
%e A302503 Terms a(1) to a(10) are obvious;
%e A302503 a(11) is 23 because 23 is the smallest integer not yet in the sequence such that the elements of the sets {3,4,5,6,7,8,9,2} and {4,5,6,7,8,9,2,3} are eight consecutive digits;
%e A302503 a(12) is 45 because 45 is the smallest integer not yet in the sequence such that the elements of the sets {5,6,7,8,9,2,3,4} and {6,7,8,9,2,3,4,5} are eight consecutive digits;
%e A302503 a(13) is 67 because 67 is the smallest integer not yet in the sequence such that the elements of the sets {7,8,9,2,3,4,5,6} and {8,9,2,3,4,5,6,7} are eight consecutive digits;
%e A302503 etc.
%o A302503 (Python)
%o A302503 a, runLength = [i for i in range(10)], 8
%o A302503 def helper(s, k, l, a):
%o A302503     if k not in a: return k
%o A302503     return min([helper(s[(2-l):]+str(i), int(str(k)+str(i)), l, a) for i in range(10) if (k!=0 or i!=0) and s.find(str(i))==-1 and (all(d[n]+1==d[n+1] for n in range(l-1)) if (d:=sorted([int((s+str(i))[n]) for n in range(l)])) else False)])
%o A302503 while len(a)<100: a.append(helper(("".join(map(str, a)))[(1-runLength):], 0, runLength, a))
%o A302503 print(a) # _Dominic McCarty_, Feb 03 2025
%Y A302503 Cf. A228326 for the same idea with sets of two digits, A302173 (sets of three digits), A302499 (sets of four digits), A302500 (sets of five digits), A302501 (sets of six digits) and A302502 (sets of seven digits).
%K A302503 nonn,base
%O A302503 1,3
%A A302503 _Eric Angelini_ and _Jean-Marc Falcoz_, Apr 09 2018