A302499 Lexicographically first sequence of distinct terms such that any set of four successive digits can be reordered as {d, d+1, d+2, d+3}, d being the smallest of the four digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 67, 85, 674, 56, 34, 52, 341, 23, 41, 230, 12, 30, 123, 45, 63, 452, 345, 234, 523, 412, 301, 2301, 2341, 2345, 634, 563, 456, 74, 567, 89, 678, 96, 78, 967, 856, 745, 6345, 6745, 6785, 6789, 67856, 785, 67456, 789, 67896, 7856, 7456, 3452, 3412, 3012, 3456, 7896

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Apr 09 2018

Keywords

• nonn
• base

Comments

As the digit 0 has no predecessor and the digit 9 has no successor here, sets of successive digits like {2,1,0,9} and {7,8,9,0} are forbidden.

Examples

			Terms a(1) to a(10) are obvious;
a(11) is 67 because 67 is the smallest integer not yet in the sequence such that the elements of the sets {7,8,9,6} and {8,9,6,7} are four consecutive digits;
a(12) is 85 because 85 is the smallest integer not yet in the sequence such that the elements of the sets {9,6,7,8} and {6,7,8,5} are four consecutive digits;
a(13) is 674 because 674 is the smallest integer not yet in the sequence such that the elements of the three sets {7,8,5,6}, {8,5,6,7} and {5,6,7,4} are four consecutive digits;
etc.
		

Links

• Dominic McCarty, Table of n, a(n) for n = 1..10000 (first 269 terms from Jean-Marc Falcoz)

Crossrefs

Cf. A228326 for the same idea with sets of two digits and A302173 for sets of three digits.

Programs

Python

a, runLength = [i for i in range(10)], 4
def helper(s, k, l, a):
    if k not in a: return k
    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)])
while len(a)<100: a.append(helper(("".join(map(str, a)))[(1-runLength):], 0, runLength, a))
print(a) # Dominic McCarty, Feb 03 2025