A273513 - OEIS

A273513

a(n) is the number of arithmetic triples n<p<q (three numbers in arithmetic progression) such that p and q contain no 2's in their ternary representation.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

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 n<k (choose the smallest k), a(n)=0, a(k)=0, a(n)=A262097(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

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).

MAPLE

f:= proc(n) option remember; local m;

m:= floor(n/3);

if n mod 3 <> 2 then procname(m)