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

1,3

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.

Links

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

Example

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.

Prog

(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

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.

Sequence in context: A069510 A062388 A070244 * A334542 A102493 A024661

Adjacent sequences: A302497 A302498 A302499 * A302501 A302502 A302503

Keyword

nonn,base,changed

Author

Eric Angelini and Jean-Marc Falcoz, Apr 09 2018

Status

approved