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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. 7
 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 (list; graph; refs; listen; history; text; internal format)
 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 Alois P. Heinz, Rows n = 0..40, flattened Eric Weisstein's World of Mathematics, Nonaveraging Sequence Wikipedia, Arithmetic progression Wikipedia, Salem-Spencer set 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

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Last modified August 9 18:32 EDT 2020. Contains 336326 sequences. (Running on oeis4.)