A359010 Variant of the inventory sequence: Record the number of terms whose value occurs once thus far in the sequence, then the number of terms whose value occurs twice thus far, and so on; a row ends when a 0 that would repeat infinitely is reached.

0, 1, 0, 1, 4, 0, 1, 0, 3, 4, 0, 1, 2, 0, 4, 0, 2, 2, 6, 4, 0, 2, 0, 0, 12, 0, 3, 2, 0, 8, 5, 0, 4, 2, 0, 4, 0, 12, 0, 3, 2, 3, 8, 0, 6, 7, 0, 2, 6, 3, 4, 5, 0, 7, 8, 0, 0, 6, 3, 8, 0, 6, 7, 8, 0, 0, 4, 3, 4, 10, 0, 7, 8, 9, 0, 2, 4, 0, 8, 5, 0, 14, 0, 9, 10, 0

Offset: 1

Author

Neal Gersh Tolunsky, Dec 11 2022

Keywords

• nonn
• tabf

Comments

Note that we are counting terms with repetition. For example, to find a(5)=4, we are looking for the number of terms that appear twice. 0 and 1 each occur twice, which is 2+2=4 (not 1+1=2). This means that each column contains only multiples of the number of occurrences it is counting.

A row ends when a 0 is reached as the k-th term in a row and the only value left occurring greater than or equal to k times is 0. - Neal Gersh Tolunsky, Feb 08 2025

Examples

			First few rows of irregular triangle:
  0;
  1,  0;
  1,  4,  0;
  1,  0,  3,  4,  0;
  1,  2,  0,  4,  0;
  2,  2,  6,  4,  0;
  2,  0,  0, 12,  0;
  3,  2,  0,  8,  5,  0;
  4,  2,  0,  4,  0, 12,  0;
  3,  2,  3,  8,  0,  6,  7,  0;
  2,  6,  3,  4,  5,  0,  7,  8,  0;
		

Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10073 (first 136 rows)

Crossrefs

Cf. A342585, A350768.

Programs

Python

from collections import Counter
from itertools import count, islice
def end_cond(I, k): # the only value left occurring >= k times is 0
    return I[0] >= k and not any(I[i] >= k for i in I if i > 0)
def agen(): # generator of terms
    I = Counter()
    while True:
        for i in count(1):
            c = sum(v for v in I.values() if v==i)
            yield c
            I[c] += 1
            if c == 0 and end_cond(I, i):
                break
print(list(islice(agen(), 86))) # Michael S. Branicky, Jan 28 2025