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 A281871 Number T(n,k) of k-element subsets of [n] having a square element sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 24
 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 2, 0, 0, 1, 2, 3, 3, 2, 1, 0, 1, 2, 4, 5, 5, 2, 1, 0, 1, 2, 5, 8, 8, 6, 3, 0, 1, 1, 3, 6, 11, 14, 13, 7, 4, 1, 0, 1, 3, 7, 15, 23, 24, 19, 10, 3, 1, 0, 1, 3, 8, 20, 34, 43, 39, 25, 13, 3, 1, 0, 1, 3, 9, 26, 49, 71, 74, 60, 34, 14, 5, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,12 LINKS Alois P. Heinz, Rows n = 0..200, flattened FORMULA T(n,n) = 1 for n in { A001108 }, T(n,n) = 0 otherwise. T(n,n-1) = 1 for n in { A214857 }, T(n,n-1) = 0 for n in { A214858 }. EXAMPLE T(7,0) = 1: {}. T(7,1) = 2: {1}, {4}. T(7,2) = 4: {1,3}, {2,7}, {3,6}, {4,5}. T(7,3) = 5: {1,2,6}, {1,3,5}, {2,3,4}, {3,6,7}, {4,5,7}. T(7,4) = 5: {1,2,6,7}, {1,3,5,7}, {1,4,5,6}, {2,3,4,7}, {2,3,5,6}. T(7,5) = 2: {1,2,3,4,6}, {3,4,5,6,7}. T(7,6) = 1: {1,2,4,5,6,7}. T(7,7) = 0. T(8,8) = 1: {1,2,3,4,5,6,7,8}. Triangle T(n,k) begins: 1; 1, 1; 1, 1, 0; 1, 1, 1, 0; 1, 2, 1, 1, 0; 1, 2, 2, 2, 0, 0; 1, 2, 3, 3, 2, 1, 0; 1, 2, 4, 5, 5, 2, 1, 0; 1, 2, 5, 8, 8, 6, 3, 0, 1; 1, 3, 6, 11, 14, 13, 7, 4, 1, 0; 1, 3, 7, 15, 23, 24, 19, 10, 3, 1, 0; 1, 3, 8, 20, 34, 43, 39, 25, 13, 3, 1, 0; 1, 3, 9, 26, 49, 71, 74, 60, 34, 14, 5, 0, 0; MAPLE b:= proc(n, s) option remember; expand(`if`(n=0, `if`(issqr(s), 1, 0), b(n-1, s)+x*b(n-1, s+n))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)): seq(T(n), n=0..16); MATHEMATICA b[n_, s_] := b[n, s] = Expand[If[n == 0, If[IntegerQ @ Sqrt[s], 1, 0], b[n - 1, s] + x*b[n - 1, s + n]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *) CROSSREFS Columns k=0-10 give: A000012, A000196, A176615, A281706, A281864, A281865, A281866, A281867, A281868, A281869, A281870. Main diagonal is characteristic function of A001108. Diagonals T(n+k,n) for k=2-10 give: A281965, A281966, A281967, A281968, A281969, A281970, A281971, A281972, A281973. Row sums give A126024. T(2n,n) gives A281872. Cf. A000217, A000290, A007318, A214857, A214858, A278339, A281994, A284249. Sequence in context: A259538 A099314 A214341 * A131334 A004602 A247418 Adjacent sequences: A281868 A281869 A281870 * A281872 A281873 A281874 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jan 31 2017 STATUS approved

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Last modified December 9 23:05 EST 2022. Contains 358710 sequences. (Running on oeis4.)