A302501 Lexicographically first sequence of distinct terms such that any set of six successive digits can be reordered as {d, d+1, d+2, d+3, d+4, d+5}, d being the smallest of the six digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 45, 67, 83, 456, 72, 34, 56, 12, 345, 61, 23, 450, 123, 4501, 234, 50, 1234, 501, 2345, 612, 3450, 12345, 672, 3456, 78, 94, 567, 89, 4567, 834, 5672, 34561, 23450, 123450, 123456, 723, 4561, 23456, 783, 45672, 34567, 894, 5678, 945, 678, 9456, 789, 45678, 94567, 8345, 6723, 45612, 34501, 234501, 234561, 234567, 8945, 6783, 456723, 456123, 45012, 345012

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 {4,3,2,1,0,9} and {5,6,7,8,9,0} are forbidden.

Examples

			Terms a(1) to a(10) are obvious;
a(11) is 45 because 45 is the smallest integer not yet in the sequence such that the elements of the sets {5,6,7,8,9,4} and {6,7,8,9,4,5} are six consecutive digits;
a(12) is 67 because 67 is the smallest integer not yet in the sequence such that the elements of the sets {7,8,9,4,5,6} and {8,9,4,5,6,7} are six consecutive digits;
a(13) is 83 because 83 is the smallest integer not yet in the sequence such that the elements of the sets {9,4,5,6,7,8} and {4,5,6,7,8,3} are six consecutive digits;
etc.
		

Links

• Dominic McCarty, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A228326 for the same idea with sets of two digits, A302173 (sets of three digits), A302499 (sets of four digits) and A302500 (sets of five digits).

Programs

Python

a, runLength = [i for i in range(10)], 6
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