A376238 Sequence obtained from A376239 by deleting the substrings of k-1 copies of k starting at index k(k-1)/2 + 1 for each k > 1.

1, 1, 2, 1, 2, 3, 3, 1, 2, 4, 4, 4, 1, 3, 5, 5, 5, 5, 3, 2, 6, 6, 6, 6, 6, 1, 2, 7, 7, 7, 7, 7, 7, 4, 4, 8, 8, 8, 8, 8, 8, 8, 4, 3, 9, 9, 9, 9, 9, 9, 9, 9, 3, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 5, 5, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 5, 5, 12

Offset

1,3

Comments

In A376239 we conjecture that the sequences S[r+1] = D S[r], starting with S[1] = A376239, will have the property P(r) for all r >= 1. Here D means to delete the subsequences of k repeated k-1 times, starting at index (k-1)r + T(k-2) + 1, for all k > 1. The property P(r) means that for all k > 1, the sequence has k-1 copies of k, starting at index (k-1)r + T(k-2) + 1, and each of these runs is preceded by r smaller terms. (Thus, r is the number of 1s preceding the first 2 in the sequence.)

With these definitions the present sequence A376238 = S[2] = D(A376239) has property P(2).

Links

Table of n, a(n) for n=1..78.

Formula

a(n) = A376239(n(n+1)/2).

Prog

(Python) A376238 = lambda n: A376239(n*(n+1)//2)
A376238_upto = lambda N: [A376238(n) for n in range(1, N+1)]
# less efficiently, for illustration:
S_upto = lambda N: [A376239(n) for n in range(1, N+1)]
def D(S, n=None):
    if not n: n=next(n for n, a in enumerate(S) if a>1)
    for c in range(1, len(S)//n): S[n*c:(n+1)*c] = []
    return S
A376238_upto = lambda N: D(S_upto(N*(N+1)//2))

Crossrefs

Cf. A376239 (initial sequence S[1]), A000217 (triangular numbers n(n+1)/2).

Sequence in context: A112531 A373553 A100002 * A348330 A328471 A227909

Adjacent sequences: A376235 A376236 A376237 * A376239 A376240 A376241

Keyword

nonn

Author

M. F. Hasler, Oct 28 2024

Status

approved