OFFSET
1,1
COMMENTS
This is the Avdispahić-Zejnulahi sequence AZ(2). For a positive integer k, the Avdispahić-Zejnulahi sequence AZ(k) is given by: a(1)=k, thereafter a(n) is the least positive integer not yet in the sequence such that Sum_{i=1..n} a(i) == k (mod n+k). It is interesting to note that (AZ(k)) represents a sequence of permutations of the set of positive integers. (See Links section for details concerning AZ(1).)
LINKS
Muharem Avdispahić and Faruk Zejnulahi, An integer sequence with a divisibility property, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.
Jeffrey Shallit, Proving properties of some greedily-defined integer recurrences via automata theory, arXiv:2308.06544 [cs.DM], 2023.
MATHEMATICA
lst={2}; f[s_List]:=Block[{k=1, len=3+Length@lst, t=Plus@@lst}, While[MemberQ[s, k]||Mod[k+t, len]!=2, k++]; AppendTo[lst, k]]; Nest[f, lst, 100]
PROG
(Python)
z_list = [-1, 2, 4]
m_list = [-1, 0, 1]
n = 2
for n in range(2, 100):
if m_list[n] in z_list:
m_list.append(m_list[n] + 1)
z_list.append(m_list[n+1] + n+2)
else:
m_list.append(m_list[n])
z_list.append(m_list[n+1])
print(z_list[1:])
CROSSREFS
KEYWORD
nonn
AUTHOR
Zenan Sabanac, Nov 03 2023
STATUS
approved