A302500 Lexicographically first sequence of distinct terms such that any set of five successive digits can be reordered as {d, d+1, d+2, d+3, d+4}, d being the smallest of the five digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 56, 78, 45, 67, 34, 562, 345, 12, 340, 123, 40, 1234, 51, 23, 401, 234, 512, 3401, 2340, 12340, 12345, 62, 3451, 2345, 623, 451, 23401, 23451, 23456, 73, 456, 734, 567, 84, 5673, 4562, 3456, 784, 5678, 95, 678, 956, 789, 56784, 56734, 5623, 4512, 34012, 34512, 34562, 34567, 89

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

Examples

			Terms a(1) to a(10) are obvious;
a(11) is 56 because 56 is the smallest integer not yet in the sequence such that the elements of the sets {6,7,8,9,5} and {7,8,9,5,6} are five consecutive digits;
a(12) is 78 because 78 is the smallest integer not yet in the sequence such that the elements of the sets {8,9,5,6,7} and {9,5,6,7,8} are five consecutive digits;
a(13) is 45 because 45 is the smallest integer not yet in the sequence such that the elements of the sets {5,6,7,8,4} and {6,7,8,4,5} are five consecutive digits;
etc.
		

Links

• Dominic McCarty, Table of n, a(n) for n = 1..10000 (first 257 terms from Jean-Marc Falcoz)

Crossrefs

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

Programs

Python

a, runLength = [i for i in range(10)], 5
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