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

0,5

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).

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

Formula

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

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

Example

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;
  ...

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 *)

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.

Sequence in context: A215064 A124054 A299208 * A082870 A026009 A137171

Adjacent sequences: A334184 A334185 A334186 * A334188 A334189 A334190

Keyword

nonn,tabf

Author

Alois P. Heinz, May 14 2020

Status

approved