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A367066 a(n) = ((Sum_{i=1..n} A367065(i))-2)/(n+2).

%I A367066 #25 Dec 16 2023 14:14:45
%S A367066 0,1,1,2,3,3,4,5,5,6,6,7,8,8,9,10,10,11,11,12,13,13,14,15,15,16,16,17,
%T A367066 18,18,19,19,20,21,21,22,23,23,24,24,25,26,26,27,28,28,29,29,30,31,31,
%U A367066 32,32,33,34,34,35,36,36,37,37,38,39,39,40,41,41,42,42,43
%N A367066 a(n) = ((Sum_{i=1..n} A367065(i))-2)/(n+2).
%C A367066 For a positive integer k define the Avdispahić-Zejnulahi sequence AZ(k) by b(1)=k, and thereafter b(n) is the least positive integer not yet in the sequence such that Sum_{i=1..n} b(i) == k (mod n+k). Define the Avdispahić-Zejnulahi means sequence AZM(k) by a(n) = ((Sum_{i=1..n} b(i))-k)/(n+k). This is the AZM(2) sequence.
%H A367066 Muharem Avdispahić and Faruk Zejnulahi, <a href="https://www.researchgate.net/publication/341726940_AN_INTEGER_SEQUENCE_WITH_A_DIVISIBILITY_PROPERTY">An integer sequence with a divisibility property</a>, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.
%F A367066 Conjecture: a(n) = floor(n/phi + 1/phi^3) - [n+2 = Fibonacci(2*j+1) for some j], where phi = (1+sqrt(5))/2 and [] is the Iverson bracket. - _Jon E. Schoenfield_, Nov 03 2023
%t A367066 zlist={-1,2,4};
%t A367066 mlist={-1,0,1};
%t A367066 For[n=3,n<=101,n++,If[MemberQ[zlist,mlist[[n]]],AppendTo[mlist,mlist[[n]]+1];
%t A367066 AppendTo[zlist,mlist[[n+1]]+n+1];,AppendTo[mlist,mlist[[n]]];AppendTo[zlist,mlist[[n+1]]];];];
%t A367066 mlist=Drop[mlist,1];mlist
%o A367066 (Python)
%o A367066 z_list=[-1,2,4]
%o A367066 m_list=[-1,0,1]
%o A367066 n=2
%o A367066 for n in range(2, 100):
%o A367066     if m_list[n] in z_list:
%o A367066         m_list.append(m_list[n] + 1)
%o A367066         z_list.append(m_list[n+1] + n+2)
%o A367066     else:
%o A367066         m_list.append(m_list[n])
%o A367066         z_list.append(m_list[n+1])
%o A367066 print(m_list[1:])
%Y A367066 Cf. A367065.
%K A367066 nonn
%O A367066 1,4
%A A367066 _Zenan Sabanac_, Nov 03 2023