OFFSET
0,3
COMMENTS
a(0) = 0, a(1) = 1, and for n >= 2, a(n) is the least integer t > a(n-1) such that for all 0 < i <= n/2 we have a(n-2i)+t <> 2a(n-i).
By double arithmetic sequence it is meant that both the indices and the values are in arithmetic progression.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
T. Brown, V. Jungic, and A. Poelstra, On double 3-term arithmetic progressions, arxiv preprint, November 2013.
EXAMPLE
a(8) = 15: 12 is not in the sequence because a(6) = 10, a(7) = 11; 13 is not in the sequence because a(4) = 7, a(6) = 10; 14 is not in the sequence because a(0) = 0, a(4) = 7, so a(8) = 15.
MAPLE
a:= proc(n) option remember; local i, t, ok;
if n<2 then n
else for t from 1+a(n-1) do ok:=true;
for i to n/2 while ok
do ok:=a(n-2*i)+t <> 2*a(n-i) od;
if ok then return t fi
od
fi
end:
seq(a(n), n=0..100); # Alois P. Heinz, May 26 2014
MATHEMATICA
a[n_] := a[n] = Module[{i, t, ok}, If[n<2, n, For[t = 1+a[n-1], True, t++, ok = True; i = 1; While[ok && i <= n/2, ok = a[n-2*i]+t != 2*a[n-i]; i++]; If[ok, Return[t]]]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 09 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jeffrey Shallit, May 26 2014
STATUS
approved