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0, 2, 7, 21, 23, 64, 69, 71, 193, 207, 209, 214, 579, 581, 622, 627, 629, 643, 1737, 1739, 1744, 1866, 1868, 1882, 1887, 1889, 1930, 5211, 5213, 5218, 5232, 5234, 5599, 5604, 5606, 5647, 5661, 5663, 5668, 5790, 5792, 15634, 15639, 15641, 15655, 15696, 15698
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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.
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FORMULA
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MAPLE
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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:
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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