OFFSET
1,2
COMMENTS
From Robert Israel, Feb 03 2016: (Start)
a(n) is the first member of the n-th sequence in the greedy partition of the nonnegative integers into sequences that contain no 3-term arithmetic progression.
As a special case (proved by Roth in 1953) of Szemerédi's theorem, sequences with no 3-term arithmetic progressions must have density 0. In particular, the nonnegative integers can't be partitioned into finitely many such sequences. Therefore this sequence is infinite.
a(n+1) >= a(n) + 2. There seem to be many cases where this is an equality. (End)
It can be deduced from the main result of Gerver, Propp, Simpson (below) that a(3n+1) = 3a(2n+1), a(3n+2) = 2 + 3a(2n+1), and a(3n) = 1 + 3a(2n). This implies infinitely many cases where a(n+1) = a(n) + 2. - C. Kenneth Fan, Dec 09 2018
Indices of records in A006997. - Rémy Sigrist, Jan 06 2024
LINKS
Robert Israel, Table of n, a(n) for n = 1..140
Matvey Borodin, Hannah Han, Kaylee Ji, Tanya Khovanova, Alexander Peng, David Sun, Isabel Tu, Jason Yang, William Yang, Kevin Zhang, Kevin Zhao, Variants of Base 3 over 2, arXiv:1901.09818 [math.NT], 2019.
J. Gerver, J. Propp, J. Simpson, Greedily partitioning the natural numbers into sets free of arithmetic progressions, Proc. of the Amer. Math. Soc. 102 (1988), no. 3, pp. 765-772.
Tanya Khovanova and Kevin Wu, Base 3/2 and Greedily Partitioned Sequences, arXiv:2007.09705 [math.NT], 2020.
K. F. Roth, On certain sets of integers, Journal of the London Mathematical Society s1-28 (1953), 104-109.
Wikipedia, Szemerédi's theorem.
FORMULA
A006997(a(n)) = n - 1. - Rémy Sigrist, Jan 06 2024
MAPLE
M:= 100: # to get a(1) to a(M)
for i from 1 to M do B[i]:= {}: F[i]:= {}: od:
for x from 0 do
for i from 1 to M do
if not member(x, F[i]) then
F[i]:= F[i] union map(y -> 2*x-y, B[i]);
B[i]:= B[i] union {x};
if not assigned(A[i]) then A[i]:= x fi;
break
fi
od;
if i = M+1 then break fi;
od:
seq(A[i], i=1..M); # Robert Israel, Feb 03 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Max Barrentine, Dec 06 2015
STATUS
approved