A176615 -id:A176615 - cp's OEIS frontend

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.

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

Offset: 0

Alois P. Heinz, Jan 31 2017

			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;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

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, A377572.

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);

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 *)

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 }.
Sum_{k=0..n} k * T(n,k) = A377572(n).

A281706 Number of sets of three positive numbers <= n whose sum is a square.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 5, 8, 11, 15, 20, 26, 33, 40, 48, 57, 67, 80, 93, 107, 121, 136, 153, 172, 193, 214, 236, 259, 284, 311, 340, 371, 402, 433, 466, 501, 538, 577, 618, 661, 705, 751, 798, 847, 897, 949, 1002, 1057, 1115, 1176, 1239, 1303, 1369, 1436, 1505

Offset: 0

Alois P. Heinz and Robert G. Wilson v, Jan 28 2017

Inspired by A278329.

a(n) >= a(n-1), first differences are nondecreasing.

			a(1) .. a(3) = 0;
a(4) = 1 since 2+3+4 = 9;
a(5) = 2 since 2+3+4 = 1+3+5 = 9;
a(6) = 3 since 2+3+4 = 1+3+5 = 1+2+6 = 9;
a(7) = 5 since a(6) = 3 plus 3+6+7 = 4+5+7 = 16;
a(8) = 8 since a(7) = 5 plus 1+7+8 = 2+6+8 = 3+5+8 = 16; etc.

Alois P. Heinz and Robert G. Wilson v, Table of n, a(n) for n = 0..20000

Cf. A176615, A278329.

Column k=3 of A281871.

g:= (n, t)-> max(0, iquo(n-1, 2)-max(1, n-t)+1):
b:= n-> add(g(k^2-n, n-1), k=ceil(sqrt(n+3))..floor(sqrt(3*n-3))):
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+b(n)) end:
seq(a(n), n=0..100);

f[n_] := Length@ Select[Plus @@@ Subsets[ Range@ n, {3}], Mod[ Sqrt[#], 1] == 0 &]; Array[f, 65] (* or *)
f = Compile[{{n, _Integer}}, Block[{a = 1, b = 2, c = 3, cnt = 0}, While[a < b, b = a +1; While[b < c, c = b +1; While[c < n +1, If[ Mod[ Sqrt[a + b + c], 1] == 0, cnt++]; c++]; b++]; a++]; cnt]]; Array[f, 65] (* or *)
g = Compile[{{n, Integer}}, If[n < 4, 0, Block[{a = 1, b = 2, cnt = f[n -1]}, While[a < b, b = a +1; While[b < n, If[ Mod[ Sqrt[a + b + n], 1] == 0, cnt++]; b++]; a++]; cnt]]]; f[n] := f[n] = g[n]; Array[f, 100]

cp's OEIS Frontend

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.

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