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A302502 Lexicographically first sequence of distinct terms such that any set of seven successive digits can be reordered as {d, d+1, d+2, d+3, d+4, d+5, d+6}, d being the smallest of the seven digits.

%I A302502 #16 Feb 03 2025 17:04:56
%S A302502 0,1,2,3,4,5,6,7,8,9,34,56,78,23,45,67,12,345,60,123,456,71,234,560,
%T A302502 1234,567,82,3456,712,34560,12345,601,2345,671,23456,782,34567,89,
%U A302502 345678,93,4567,823,45671,234560,123456,789,3456782,345671,234567,893,45678,934,5678,9345,678,93456,7823,456712,345601
%N A302502 Lexicographically first sequence of distinct terms such that any set of seven successive digits can be reordered as {d, d+1, d+2, d+3, d+4, d+5, d+6}, d being the smallest of the seven digits.
%C A302502 As the digit 0 has no predecessor and the digit 9 has no successor here, sets of successive digits like {5,4,3,2,1,0,9} and {4,5,6,7,8,9,0} are forbidden.
%H A302502 Dominic McCarty, <a href="/A302502/b302502.txt">Table of n, a(n) for n = 1..10000</a>
%e A302502 Terms a(1) to a(10) are obvious;
%e A302502 a(11) is 34 because 34 is the smallest integer not yet in the sequence such that the elements of the sets {4,5,6,7,8,9,3} and {5,6,7,8,9,3,4} are seven consecutive digits;
%e A302502 a(12) is 56 because 56 is the smallest integer not yet in the sequence such that the elements of the sets {6,7,8,9,3,4,5} and {7,8,9,3,4,5,6} are seven consecutive digits;
%e A302502 a(13) is 78 because 78 is the smallest integer not yet in the sequence such that the elements of the sets {8,9,3,4,5,6,7} and {9,3,4,5,6,7,8} are seven consecutive digits;
%e A302502 etc.
%o A302502 (Python)
%o A302502 a, runLength = [i for i in range(10)], 7
%o A302502 def helper(s, k, l, a):
%o A302502     if k not in a: return k
%o A302502     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 A302502 while len(a)<100: a.append(helper(("".join(map(str, a)))[(1-runLength):], 0, runLength, a))
%o A302502 print(a) # _Dominic McCarty_, Feb 03 2025
%Y A302502 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) and A302501 (sets of six digits).
%K A302502 nonn,base
%O A302502 1,3
%A A302502 _Eric Angelini_ and _Jean-Marc Falcoz_, Apr 09 2018