A378821 Lexicographically earliest sequence of distinct positive integers such that the count of integers between a(n) and a(n-1), excluding values already in the sequence, is distinct from the same count for any other a(k) and a(k-1) at the time they occurred.

1, 2, 4, 7, 11, 3, 10, 18, 5, 16, 26, 6, 22, 35, 8, 27, 42, 9, 32, 51, 12, 37, 58, 13, 41, 66, 14, 47, 74, 15, 53, 83, 17, 57, 90, 19, 62, 98, 20, 68, 106, 21, 73, 115, 23, 78, 122, 24, 84, 131, 25, 88, 138, 28, 94, 147, 29, 99, 154, 30, 103, 162, 31, 109, 170, 33

Offset: 1

Author

Neal Gersh Tolunsky, Dec 08 2024

Keywords

• nonn

Examples

			 a(1-5) = 1,2,4,7,11 follow a straightforward pattern of counting up, where each pair of consecutive terms encloses (in order) 0,1,2,3 unused values.
  1,2,3,4,5,6,7,8,9,10,11
  1 2 * 4 * * 7 * * *  11
  ^ ^   ^     ^        ^
  At a(6), a(5)=11 and a(6)=3 enclose 5 unused values:
  1,2,3,4,5,6,7,8,9,10,11
  1 2 3 4 * * 7 * * *  11
      ^                ^
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A378822, A081145, A091263.

Programs

Python

from itertools import count, islice
def c(k, m, a): return sum(1 for i in range(min(k, m)+1, max(k, m)) if i not in a)
def agen(): # generator of terms
    a, d, an, m = set(), set(), 1, 2
    while True:
        yield an
        a.add(an)
        found, k = False, m
        if m < an:
            ck = c(k, an, a)
            for k in range(m, an):
                if k not in a:
                    if ck not in d:
                        found = True
                        break
                    ck -= 1
        if not found:
            kk = max(m, an+1)
            ck = c(kk, an, a)
            for k in count(kk):
                if k not in a:
                    if ck not in d:
                        found = True
                        break
                    ck += 1
        d.add(c(an, k, a))
        an = k
        while m in a: m += 1
print(list(islice(agen(), 66))) # Michael S. Branicky, Dec 09 2024