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.

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, 44, 25, 1, 9, 36, 68, 51, 4, 1, 10, 45, 100, 98, 24, 1, 11, 55, 140, 165, 64, 7, 1, 12, 66, 190, 267, 144, 25, 1, 13, 78, 250, 407, 284, 78, 6, 1, 14, 91, 322, 601, 520, 188, 22, 1

Offset: 0

Author

Alois P. Heinz, May 14 2020

Keywords

• nonn
• tabf

Comments

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).

Examples

			Triangle T(n,k) begins:
  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,  44,   25;
  1,  9,  36,  68,   51,    4;
  1, 10,  45, 100,   98,   24;
  1, 11,  55, 140,  165,   64,   7;
  1, 12,  66, 190,  267,  144,  25;
  1, 13,  78, 250,  407,  284,  78,   6;
  1, 14,  91, 322,  601,  520, 188,  22,  1;
  1, 15, 105, 406,  849,  862, 386,  64,  4;
  1, 16, 120, 504, 1175, 1394, 763, 164, 14;
  ...
		

Links

• Fausto A. C. Cariboni, Rows n = 0..70, flattened (rows n = 0..40 from Alois P. Heinz)
• Eric Weisstein's World of Mathematics, Nonaveraging Sequence
• Wikipedia, Arithmetic progression
• Wikipedia, Salem-Spencer set
• Index entries related to non-averaging sequences

Crossrefs

Columns k=0-4 give: A000012, A000027, A000217(n-1), A212964(n-1), A300760.

Row sums give A051013.

Last elements of rows give A262347.

Cf. A003002, A334892.

Programs

Maple

b:= proc(n, s) option remember; `if`(n=0, 1, b(n-1, s)+ `if`(
      ormap(j-> 2*j-n in s, s), 0, expand(x*b(n-1, s union {n}))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, {})):
seq(T(n), n=0..16);

Mathematica

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[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, {}]];
T /@ Range[0, 16] // Flatten (* Jean-François Alcover, May 30 2020, after Maple *)

Formula

T(n,k) = Sum_{j=0..n} A334892(j,k).

T(n,A003002(n)) = A262347(n).