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.
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).
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, ...
(for row lengths see A003945)
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 *)
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
print(aupton(76)) # Michael S. Branicky, Sep 19 2021
CROSSREFS
KEYWORD
AUTHOR
David James Sycamore, Sep 12 2021
EXTENSIONS
Offset changed to 0 by N. J. A. Sloane, Sep 12 2021
STATUS
approved