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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.
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%I #49 Sep 19 2021 13:03:21

%S 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,

%T 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,

%U 17,16,16,17,18,18,17,15,13,11,10,7,6,5,4,4,4,4,3,3

%N 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.

%C 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.

%C 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).

%H Michael De Vlieger, <a href="/A347738/b347738.txt">Table of n, a(n) for n = 0..24573</a> (rows 0 <= k <= 12 when considered as an irregular triangle)

%H Michael De Vlieger, <a href="/A347738/a347738.png">log-log scatterplot of a(n)</a> for 1 <= n <= 49150 (ignoring zeros).

%e As an irregular triangle this begins:

%e 0;

%e 1, 1, 0;

%e 4, 3, 2, 2, 1, 0;

%e 10, 8, 6, 5, 5, 5, 3, 2, 2, 1, 1, 0;

%e 22, 19, 15, 12, 11, 11, 9, 9, 10, 10, 9, 6, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 0;

%e 46, ...

%e (for row lengths see A003945)

%t 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 *)

%o (Python)

%o def aupton(nn):

%o num, gte_inventory, alst, bigc = 0, [1], [0], 0

%o while len(alst) < nn+1:

%o c = gte_inventory[num] if num <= bigc else 0

%o num = 0 if c == 0 else num + 1

%o for i in range(min(c, bigc)+1):

%o gte_inventory[i] += 1

%o for i in range(bigc+1, c+1):

%o gte_inventory.append(1)

%o bigc = len(gte_inventory) - 1

%o alst.append(c)

%o return alst

%o print(aupton(76)) # _Michael S. Branicky_, Sep 19 2021

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

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

%K nonn,tabf,nice,look

%O 0,5

%A _David James Sycamore_, Sep 12 2021

%E Offset changed to 0 by _N. J. A. Sloane_, Sep 12 2021