A358066 Inventory sequence: record where the 1's, 2's, etc. are located starting with a(1) = 1, a(2) = 1 (see example).

1, 1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 5, 4, 6, 7, 1, 2, 3, 5, 9, 4, 6, 10, 7, 11, 8, 13, 1, 2, 3, 5, 9, 16, 4, 6, 10, 17, 7, 11, 18, 8, 13, 21, 12, 19, 1, 2, 3, 5, 9, 16, 28, 4, 6, 10, 17, 29, 7, 11, 18, 30, 8, 13, 21, 34, 12, 19, 31, 14, 22, 35, 1, 2, 3, 5, 9, 16, 28, 46, 4, 6, 10, 17, 29, 47, 7, 11, 18, 30, 48

Offset

1,4

Links

Table of n, a(n) for n=1..90.

Example

At stage n >= 1 we only look at the numbers 1 up to n, and ignore numbers bigger than n.
Stage 0: start with a(1) = 1, a(2) = 1.
Stage 1: we see 1's at 1,2, so we adjoin 1,2, getting 1,1, 1,2.
Stage 2: we see 1's at 1,2,3, and 2's at 4, so we adjoin 1,2,3,4, getting 1,1,1,2, 1,2,3,4.
Stage 3: we see 1's at 1,2,3,5, 2's at 4,6 and 3's at 7, so we adjoin 1,2,3,5,4,6,7, getting 1,1,1,2,1,2,3,4, 1,2,3,5,4,6,7.
Stage 4: we see 1's at 1,2,3,5,9, 2's at 4,6,10, 3's at 7,11, 4's at 8,13, so we adjoin 1,2, ..., 8,13 and so on.
We obtain an irregular triangle by writing the results of the stages as separate rows:
1, 1,
1, 2,
1, 2, 3, 4,
1, 2, 3, 5, 4, 6, 7,
1, 2, 3, 5, 9, 4, 6, 10, 7, 11, 8, 13,
1, 2, 3, 5, 9, 16, 4, 6, 10, 17, 7, 11, 18, 8, 13, 21, 12, 19,
1, 2, 3, 5, 9, 16, 28, 4, 6, 10, 17, 29, 7, 11, 18, 30, 8, 13, 21, 34, 12, 19, 31, 14, 22, 35,
... (N. J. A. Sloane, Nov 07 2022)

Prog

(Python)
terms = [1, 1]
for i in range(1, 11):
    new_terms = []
    for j in range(1, i+1):
        for k in range(len(terms)):
            if terms[k] == j: new_terms.append(k+1)
    terms.extend(new_terms)
print(terms) # Gleb Ivanov, Nov 01 2022

Crossrefs

See A357443 for another version.

Cf. A030717, A055187, A217780, A342585, A357317, A356784.

Sequence in context: A168265 A062050 A358086 * A358085 A357443 A233782

Adjacent sequences: A358063 A358064 A358065 * A358067 A358068 A358069

Keyword

nonn,tabf

Author

Ctibor O. Zizka, Oct 29 2022

Status

approved