A302504 Lexicographically first sequence of distinct terms such that any set of nine successive digits can be reordered as {d, d+1, d+2, d+3, d+4, d+5, d+6, d+7, d+8}, d being the smallest of the nine digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 34, 56, 78, 91, 23, 45, 67, 80, 123, 456, 780, 1234, 567, 89, 12345, 678, 912, 345, 6780, 123456, 789, 1234567, 801, 234, 5678, 9123, 4567, 891, 2345, 6789, 12345678, 91234, 56780, 123456780, 123456789

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Apr 09 2018

Keywords

• nonn
• base
• look

Comments

As the digit 0 has no predecessor and the digit 9 has no successor here, sets of successive digits like {7,6,5,4,3,2,1,0,9} and {2,3,4,5,6,7,8,9,0} are forbidden.

Examples

			Terms a(1) to a(10) are obvious;
a(11) is 12 because 12 is the smallest integer not yet in the sequence such that the elements of the sets {2,3,4,5,6,7,8,9,1} and {3,4,5,6,7,8,9,1,2} are nine consecutive digits;
a(12) is 34 because 34 is the smallest integer not yet in the sequence such that the elements of the sets {4,5,6,7,8,9,1,2,3} and {5,6,7,8,9,1,2,3,4} are nine consecutive digits;
a(13) is 56 because 56 is the smallest integer not yet in the sequence such that the elements of the sets {6,7,8,9,1,2,3,4,5} and {7,8,9,1,2,3,4,5,6} are nine 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), A302500 (sets of five digits), A302501 (sets of six digits), A302502 (sets of seven digits) and A302503 (sets of eight digits).

Programs

Python

a, runLength = [i for i in range(10)], 9
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 02 2025