A367065 -id:A367065 - cp's OEIS frontend

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

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, 18, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 32, 33, 34, 34, 35, 36, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 42, 43

Offset: 1

Zenan Sabanac, Nov 03 2023

nonn

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.

Muharem Avdispahić and Faruk Zejnulahi, An integer sequence with a divisibility property, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.

Cf. A367065.

zlist={-1,2,4};
mlist={-1,0,1};
For[n=3,n<=101,n++,If[MemberQ[zlist,mlist[[n]]],AppendTo[mlist,mlist[[n]]+1];
AppendTo[zlist,mlist[[n+1]]+n+1];,AppendTo[mlist,mlist[[n]]];AppendTo[zlist,mlist[[n+1]]];];];
mlist=Drop[mlist,1];mlist

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(m_list[1:])

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

A367067 a(1)=3, thereafter a(n) is the least positive integer not yet in the sequence such that Sum_{i=1..n} a(i) == 3 (mod n+3).

Original entry on oeis.org

3, 5, 1, 8, 2, 11, 13, 4, 16, 18, 6, 21, 7, 24, 26, 9, 29, 10, 32, 34, 12, 37, 39, 14, 42, 15, 45, 47, 17, 50, 52, 19, 55, 20, 58, 60, 22, 63, 23, 66, 68, 25, 71, 73, 27, 76, 28, 79, 81, 30, 84, 31, 87, 89, 33, 92, 94, 35, 97, 36, 100, 102, 38, 105

Offset: 1

Zenan Sabanac, Nov 03 2023

nonn

This is the Avdispahić-Zejnulahi sequence AZ(3).

Note that AZ(3) is the third term in a sequence of permutations of the set of positive integers defined by a specific divisibility property (see Links section and Crossrefs for details).

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.

Cf. A340510, A367065.

lst = {3};
f[s_List] := Block[{k = 1, len = 4 + Length@lst, t = Plus @@ lst},
  While[MemberQ[s, k] || Mod[k + t, len] != 3, k++];
  AppendTo[lst, k]]; Nest[f, lst, 100]

z_list=[-1,3,5]
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+3)
    else:
        m_list.append(m_list[n])
        z_list.append(m_list[n+1])
print(z_list[1:])

cp's OEIS Frontend

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

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