A273513 a(n) is the number of arithmetic triples n

0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 2, 3, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 2, 1, 1, 4, 3, 3, 5, 2, 2, 4, 2, 2, 5, 3, 3, 4, 0, 0, 1, 0, 0, 1

Offset: 0

Author

Max Barrentine, May 23 2016

Keywords

• nonn
• base
• look

Comments

This is a recursive sequence that gives the number of times n is rejected from A005836, if n is the smallest member of an arithmetic triple whose final two terms are contained in A005836.

This is similar to both A002487, which has a similar recurrence relation and counts hyperbinary representations of n, and A000119, which counts representations of n as a sum of distinct Fibonacci numbers.

For nA262097(k), a(n+1)=A262097(k-1), a(n+2)=A262097(k-2)... a(k)=A262097(n).

Indices of maxima between a(n) and a(k) appear to converge to (3/4)(k-n) and (11/12)(k-n).

Links

• Max Barrentine, Table of n, a(n) for n = 0..19683 (terms 1 through 10000 from Robert Israel)
• Robert Israel, Plot of first 10^5 terms

Crossrefs

Cf. A000119, A002487, A005836, A262097, A273514.

Programs

Maple

f:= proc(n) option remember; local m;
m:= floor(n/3);
if n mod 3 <> 2 then procname(m)
elif procname(m+1)=0 then 1 + procname(m)
else procname(m) + procname(m+1)
fi
end proc:
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Jun 16 2016

Formula

a(0)=0, a(n)=a(3n)=a(3n+1);

if a(n+1)=0, a(3n+2)=1+a(n), otherwise a(3n+2)=a(n)+a(n+1).