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A334187 Number T(n,k) of k-element subsets of [n] avoiding 3-term arithmetic progressions; triangle T(n,k), n>=0, 0<=k<=A003002(n), read by rows.

%I A334187 #53 Feb 16 2025 08:34:00
%S A334187 1,1,1,1,2,1,1,3,3,1,4,6,2,1,5,10,6,1,1,6,15,14,4,1,7,21,26,10,1,8,28,
%T A334187 44,25,1,9,36,68,51,4,1,10,45,100,98,24,1,11,55,140,165,64,7,1,12,66,
%U A334187 190,267,144,25,1,13,78,250,407,284,78,6,1,14,91,322,601,520,188,22,1
%N A334187 Number T(n,k) of k-element subsets of [n] avoiding 3-term arithmetic progressions; triangle T(n,k), n>=0, 0<=k<=A003002(n), read by rows.
%C A334187 T(n,k) is defined for all n >= 0 and k >= 0.  The triangle contains only elements with 0 <= k <= A003002(n).  T(n,k) = 0 for k > A003002(n).
%H A334187 Fausto A. C. Cariboni, <a href="/A334187/b334187.txt">Rows n = 0..70, flattened</a> (rows n = 0..40 from Alois P. Heinz)
%H A334187 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NonaveragingSequence.html">Nonaveraging Sequence</a>
%H A334187 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>
%H A334187 Wikipedia, <a href="https://en.wikipedia.org/wiki/Salem-Spencer_set">Salem-Spencer set</a>
%H A334187 <a href="/index/No#non_averaging">Index entries related to non-averaging sequences</a>
%F A334187 T(n,k) = Sum_{j=0..n} A334892(j,k).
%F A334187 T(n,A003002(n)) = A262347(n).
%e A334187 Triangle T(n,k) begins:
%e A334187   1;
%e A334187   1,  1;
%e A334187   1,  2,   1;
%e A334187   1,  3,   3;
%e A334187   1,  4,   6,   2;
%e A334187   1,  5,  10,   6,    1;
%e A334187   1,  6,  15,  14,    4;
%e A334187   1,  7,  21,  26,   10;
%e A334187   1,  8,  28,  44,   25;
%e A334187   1,  9,  36,  68,   51,    4;
%e A334187   1, 10,  45, 100,   98,   24;
%e A334187   1, 11,  55, 140,  165,   64,   7;
%e A334187   1, 12,  66, 190,  267,  144,  25;
%e A334187   1, 13,  78, 250,  407,  284,  78,   6;
%e A334187   1, 14,  91, 322,  601,  520, 188,  22,  1;
%e A334187   1, 15, 105, 406,  849,  862, 386,  64,  4;
%e A334187   1, 16, 120, 504, 1175, 1394, 763, 164, 14;
%e A334187   ...
%p A334187 b:= proc(n, s) option remember; `if`(n=0, 1, b(n-1, s)+ `if`(
%p A334187       ormap(j-> 2*j-n in s, s), 0, expand(x*b(n-1, s union {n}))))
%p A334187     end:
%p A334187 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, {})):
%p A334187 seq(T(n), n=0..16);
%t A334187 b[n_, s_] := b[n, s] = If[n == 0, 1, b[n-1, s] + If[AnyTrue[s, MemberQ[s, 2 # - n]&], 0, Expand[x b[n-1, s ~Union~ {n}]]]];
%t A334187 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, {}]];
%t A334187 T /@ Range[0, 16] // Flatten (* _Jean-François Alcover_, May 30 2020, after Maple *)
%Y A334187 Columns k=0-4 give: A000012, A000027, A000217(n-1), A212964(n-1), A300760.
%Y A334187 Row sums give A051013.
%Y A334187 Last elements of rows give A262347.
%Y A334187 Cf. A003002, A334892.
%K A334187 nonn,tabf
%O A334187 0,5
%A A334187 _Alois P. Heinz_, May 14 2020