A376238 - OEIS

%I #5 Oct 31 2024 01:02:55

%S 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,

%T 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,

%U 5,11,11,11,11,11,11,11,11,11,11,5,5,12

%N 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.

%C 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.)

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

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

%o (Python) A376238 = lambda n: A376239(n*(n+1)//2)

%o A376238_upto = lambda N: [A376238(n) for n in range(1,N+1)]

%o # less efficiently, for illustration:

%o S_upto = lambda N: [A376239(n) for n in range(1,N+1)]

%o def D(S, n=None):

%o     if not n: n=next(n for n,a in enumerate(S) if a>1)

%o     for c in range(1,len(S)//n): S[n*c:(n+1)*c] = []

%o     return S

%o A376238_upto = lambda N: D(S_upto(N*(N+1)//2))

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

%K nonn

%O 1,3

%A _M. F. Hasler_, Oct 28 2024