A361287 A variant of the inventory sequence A342585: now a row ends when the number of occurrences of the largest term in the sequence thus far has been recorded.

0, 1, 1, 1, 3, 0, 1, 2, 4, 1, 1, 1, 2, 7, 2, 1, 1, 0, 0, 1, 4, 10, 3, 2, 2, 0, 0, 1, 0, 0, 1, 8, 12, 5, 2, 2, 1, 0, 1, 1, 0, 1, 0, 1, 11, 17, 7, 2, 2, 1, 0, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 17, 23, 10, 2, 2, 1, 0, 2, 1, 0, 2, 1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0

Offset

0,5

Links

Samuel Harkness, Table of n, a(n) for n = 0..12000

Formula

a(n) is the number of all a(k) == b(n) 0 <= k < n and b(0) = 0 and b(n) = b(n-1)+1 if an a(k) > b(n) exists for 0 <= k < n, and 0 otherwise.

Example

The triangle begins:
  0;
  1, 1;
  1, 3, 0, 1;
  2, 4, 1, 1, 1;
  2, 7, 2, 1, 1, 0, 0, 1;
  4, 10, 3, 2, 2, 0, 0, 1, 0, 0, 1;
  8, 12, 5, 2, 2, 1, 0, 1, 1, 0, 1, 0, 1;
 11, 17, 7, 2, 2, 1, 0, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1;
 17, 23, 10, 2, 2, 1, 0, 2, 1, 0, 2, 1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1;
...
a(0)=0 because we begin each row by recording the number of zeros in the sequence. As yet, there are none. 0 is still the largest term in the sequence, so we begin the next row.
a(1)=1 because there is one 0. a(2)=1 because there is one 1. 1 is the largest term thus far, so we begin the next row.
a(3)=1 (there is still one zero); a(4)=3 (there are three 1s); a(5)=0 (no 2s); the row ends with a(6)=1 because there is one 3 and three is the largest term thus far.

Mathematica

len = 84; K = {0}; While[Length@K <= len, i = 0; While[i <= Max@K, AppendTo[K, Count[K, i]]; i++]]; Print[Take[K, len + 1]] (* Samuel Harkness, Mar 12 2023 *)

Prog

(Python)
from collections import Counter
from itertools import count, islice
def agen(): # generator of terms
    an, b, m, inventory = 0, 0, 0, Counter([0])
    for n in count(1):
        yield an
        an = inventory[b]
        m = max(m, an)
        b = b + 1 if m > b else 0; inventory.update([an])
print(list(islice(agen(), 85))) # Michael S. Branicky, Mar 07 2023

Crossrefs

Cf. A342585, A347317.

Sequence in context: A049765 A343394 A194801 * A273901 A014573 A067166

Adjacent sequences: A361284 A361285 A361286 * A361288 A361289 A361290

Keyword

nonn,tabf

Author

Robert Dober, Mar 07 2023

Extensions

More terms from Alois P. Heinz, Mar 07 2023

Status

approved