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A262347
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Number of subsets of [1..n] of maximal size that are free of 3-term arithmetic progressions.
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3
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1, 1, 1, 3, 2, 1, 4, 10, 25, 4, 24, 7, 25, 6, 1, 4, 14, 43, 97, 220, 2, 18, 62, 232, 2, 33, 2, 12, 36, 106, 1, 11, 2, 4, 14, 40, 2, 4, 86, 307, 20, 1, 4, 14, 41, 99, 266, 674, 1505, 3510, 7726, 14, 50, 156, 2, 8, 26, 56, 2, 4, 6, 14, 48, 2, 4, 8, 16, 28, 108, 319, 1046, 4, 26, 82, 1, 2
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OFFSET
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0,4
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COMMENTS
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The sequence A003002 gives the size of the largest subset of the integers up to n that avoids three-term arithmetic progressions. This sequence gives the number of distinct subsets of [1..n] that have that size and are free of three-term arithmetic progressions.
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LINKS
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EXAMPLE
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The largest subset of [1,6] that doesn't have any 3 terms in arithmetic progression has size 4. There are 4 such subsets with this property: {1,2,4,5}, {1,2,5,6}, {1,3,4,6} and {2,3,5,6}, so a(6)=4.
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MAPLE
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G:= proc(n, cons, t)
option remember;
local consn, consr;
if n < t or member({}, cons) then return {} fi;
if t = 0 then return {{}} fi;
consn, consr:= selectremove(has, cons, n);
consn:= subs(n=NULL, consn);
procname(n-1, consr, t) union
map(`union`, procname(n-1, consr union consn, t-1), {n});
end proc:
F:= proc(n)
local m, cons, R;
cons:= {seq(seq({i, i+j, i+2*j}, i=1..n-2*j), j=1..(n-1)/2)};
R:= G(n, cons, m+1);
if R = {} then
G(n, cons, m);
else
R
fi
end proc:
a[1]:= 1:
for n from 2 to 40 do
a[n]:= nops(F(n))
od:
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MATHEMATICA
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A003002 = Cases[Import["https://oeis.org/A003002/b003002.txt", "Table"], {_, _}][[All, 2]];
a[n_] := a[n] = Count[Subsets[Range[n], {A003002[[n+1]]}], s_ /; !MatchQ[s, {___, n1_, ___, n2_, ___, n3_, ___} /; n2 - n1 == n3 - n2]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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