OFFSET
1,3
COMMENTS
Similar to A342585, except that when we take inventory, we write down what we are counting as a subscript on the count. So if we have found k copies of m so far, we write down k_m, and include both the k and m values when we next take inventory.
More than the usual number of terms are shown, in order to match A355917.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..10020
Rémy Sigrist, PARI program
N. J. A. Sloane, The first eight inventories, with better alignment.
EXAMPLE
Initially we have no 0's, so the first inventory is 0_0. Just as in A342585, when we reach a count of zero, we take a new inventory.
Now we see two 0's, so we write down 2_0, followed by 0_1, since there are no 1's.
So the first two inventories are
0_0,
2_0, 0_1.
Now we see four 0's, so the next inventory starts 4_0, then 1_1, 1_2, and 0_3:
4_0, 1_1, 1_2, 0_3.
The first eight inventories are:
0_0,
2_0, 0_1,
4_0, 1_1, 1_2, 0_3,
6_0, 4_1, 2_2, 1_3, 2_4, 0_5,
8_0, 6_1, 5_2, 2_3, 3_4, 2_5, 2_6, 0_7,
10_0, 7_1, 9_2, 4_3, 5_4, 4_5, 3_6, 2_7, 1_8, 1_9, 1_10, 0_11,
12_0, 11_1, 11_2, 6_3, 7_4, 5_5, 5_6, 4_7, 2_8, 2_9, 2_10, 3_11, 1_12, 0_13,
14_0, 13_1, 15_2, 8_3, 9_4, 8_5, 6_6, 5_7, 5_8, 4_9, 3_10, 4_11, 2_12, 2_13, 1_14, 1_15, 0_16,
...
The sequence is obtained by reading the inventories, with each count followed by its index: 0, 0, 2, 0, 0, 1, 4, 0, 1, 1, 1, 2, 0, 3, ...
MATHEMATICA
nn = 9; c[_] = 0; a[1] = a[2] = 0; c[0] = 2; i = 3; Do[k = 0; While[c[k] > 0, Set[{a[i], a[i + 1]}, {c[k], k}]; c[a[i]]++; c[a[i + 1]]++; i += 2; k++]; Set[{a[i], a[i + 1]}, {c[k], k}]; c[a[i]]++; c[a[i + 1]]++; i += 2, {n, 2, nn}]; Array[a, i - 1] (* Michael De Vlieger, Sep 25 2022 *)
PROG
(PARI) See Links section.
(Python)
from collections import Counter
def aupton(terms):
num, alst, inventory = 0, [0, 0], Counter([0, 0])
for n in range(3, 3+terms//2):
c = [inventory[num], num]
num = 0 if c[0] == 0 else num + 1
alst.extend(c)
inventory.update(c)
return alst[:terms]
print(aupton(128)) # Michael S. Branicky, Sep 25 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Sep 24 2022
STATUS
approved