

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.


8



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,5


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).


LINKS



EXAMPLE

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, ...


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 JeanFrançois Alcover at A342585 *)


PROG

(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


CROSSREFS

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


KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



