A367066 - OEIS

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

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OFFSET

1,4

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..70.

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

FORMULA

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

MATHEMATICA

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

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(m_list[1:])

CROSSREFS

Cf. A367065.

Sequence in context: A086335 A123387 A123070 * A057361 A136409 A039729

Adjacent sequences: A367063 A367064 A367065 * A367067 A367068 A367069

KEYWORD

nonn

AUTHOR

Zenan Sabanac, Nov 03 2023

STATUS

approved