A347738 A variant of the inventory sequence: record the number of terms >= 0 thus far in the sequence, then the number of terms >= 1 thus far, then the number of terms >= 2 thus far, and so on, until a zero is recorded; the inventory then starts again, recording the number of terms >= 0, etc.

0, 1, 1, 0, 4, 3, 2, 2, 1, 0, 10, 8, 6, 5, 5, 5, 3, 2, 2, 1, 1, 0, 22, 19, 15, 12, 11, 11, 9, 9, 10, 10, 9, 6, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 0, 46, 42, 35, 28, 24, 23, 21, 20, 21, 21, 19, 17, 16, 16, 17, 18, 18, 17, 15, 13, 11, 10, 7, 6, 5, 4, 4, 4, 4, 3, 3

Offset: 0

Author

David James Sycamore, Sep 12 2021

Keywords

• nonn
• tabf
• nice
• look

Comments

Sequence starts off as A342585 but diverges after a(4). The effect is to introduce some numbers earlier in this sequence than in the original, and to stretch out the incidences of zero terms by the fact that the term immediately following a zero is now the total number of prior terms, rather than the total number of prior zero terms.

In A342585 zeros occur at positions 1,4,8,14,20,28,... (see A343880) whereas in this version they occur at positions 1,4,10,22,46,... (which is A033484, as is easily proved by induction).

Examples

			As an irregular triangle this begins:
   0;
   1,  1,  0;
   4,  3,  2,  2,  1,  0;
  10,  8,  6,  5,  5,  5, 3, 2,  2,  1, 1, 0;
  22, 19, 15, 12, 11, 11, 9, 9, 10, 10, 9, 6, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 0;
  46, ...
(for row lengths see A003945)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..24573 (rows 0 <= k <= 12 when considered as an irregular triangle)
• Michael De Vlieger, log-log scatterplot of a(n) for 1 <= n <= 49150 (ignoring zeros).

Crossrefs

Cf: A342585, A033484, A003945, A343880, A003945 (row lengths), A347324 (row sums).

A347326 has a version of this in which the rows have been normalized.

Programs

Mathematica

a[n_] := a[n] = Block[{t}, t = If[a[n - 1] == 0, 0, b[n - 1] + 1]; b[n] = t; Sum[If[a[j] >= t, 1, 0], {j, n - 1}]]; b[1] = a[1] = 0; Array[a, 77] (* Michael De Vlieger, Sep 12 2021, after Jean-François Alcover at A342585 *)

Python

def aupton(nn):
    num, gte_inventory, alst, bigc = 0, [1], [0], 0
    while len(alst) < nn+1:
        c = gte_inventory[num] if num <= bigc else 0
        num = 0 if c == 0 else num + 1
        for i in range(min(c, bigc)+1):
            gte_inventory[i] += 1
        for i in range(bigc+1, c+1):
            gte_inventory.append(1)
        bigc = len(gte_inventory) - 1
        alst.append(c)
    return alst
print(aupton(76)) # Michael S. Branicky, Sep 19 2021

Extensions

Offset changed to 0 by N. J. A. Sloane, Sep 12 2021