A334892 Number T(n,k) of k-element subsets of [n] avoiding 3-term arithmetic progressions and containing n if n>0; triangle T(n,k), n>=0, 0<=k<=A003002(n), read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 3, 2, 0, 1, 4, 4, 1, 0, 1, 5, 8, 3, 0, 1, 6, 12, 6, 0, 1, 7, 18, 15, 0, 1, 8, 24, 26, 4, 0, 1, 9, 32, 47, 20, 0, 1, 10, 40, 67, 40, 7, 0, 1, 11, 50, 102, 80, 18, 0, 1, 12, 60, 140, 140, 53, 6, 0, 1, 13, 72, 194, 236, 110, 16, 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

			  1;
  0, 1;
  0, 1,  1;
  0, 1,  2;
  0, 1,  3,  2;
  0, 1,  4,  4,   1;
  0, 1,  5,  8,   3;
  0, 1,  6, 12,   6;
  0, 1,  7, 18,  15;
  0, 1,  8, 24,  26,   4;
  0, 1,  9, 32,  47,  20;
  0, 1, 10, 40,  67,  40,   7;
  0, 1, 11, 50, 102,  80,  18;
  0, 1, 12, 60, 140, 140,  53,   6;
  0, 1, 13, 72, 194, 236, 110,  16,  1;
  0, 1, 14, 84, 248, 342, 198,  42,  3;
  0, 1, 15, 98, 326, 532, 377, 100, 10;
  ...
		

Links

• Fausto A. C. Cariboni, Rows n = 0..80, 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=0-3 give: A000007, A057427, A000027(n-1), A007590(n-2).

Row sums give A334893.

Last elements of rows give A334894.

Cf. A003002, A334187.

Programs

Maple

b:= proc(n, s) option remember; `if`(n=0, x, 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)))(
            `if`(n=0, 1, b(n-1, {n}))):
seq(T(n), n=0..16);

Mathematica

b[n_, s_] := b[n, s] = If[n == 0, x, b[n-1, s] + If[
     AnyTrue[s, MemberQ[s, 2#-n]&], 0, Expand[x*b[n-1, s ~Union~ {n}]]]];
T[n_] := If[n == 0, {1}, CoefficientList[b[n-1, {n}], x]];
T /@ Range[0, 16] // Flatten (* Jean-François Alcover, May 03 2021, after Alois P. Heinz *)

Formula

T(0,k) = A334187(0,k), T(n,k) = A334187(n,k) - A334187(n-1,k) for n > 0.