A343037 Triangle T(n,k), n >= 2, 1 <= k <= n-1, read by rows, where T(n,k) is the difference between smallest square >= binomial(n,k) and binomial(n,k).

2, 1, 1, 0, 3, 0, 4, 6, 6, 4, 3, 1, 5, 1, 3, 2, 4, 1, 1, 4, 2, 1, 8, 8, 11, 8, 8, 1, 0, 0, 16, 18, 18, 16, 0, 0, 6, 4, 1, 15, 4, 15, 1, 4, 6, 5, 9, 4, 31, 22, 22, 31, 4, 9, 5, 4, 15, 5, 34, 49, 37, 49, 34, 5, 15, 4, 3, 3, 3, 14, 9, 48, 48, 9, 14, 3, 3, 3, 2, 9, 36, 23, 23, 22, 49, 22, 23, 23, 36, 9, 2

Offset: 2

Author

Seiichi Manyama, Apr 03 2021

Keywords

• nonn
• tabl

Examples

			binomial(50,3) = binomial(50,47) = 140^2. So T(50,3) = T(50,47) = 0.
Triangle begins:
  2;
  1,  1;
  0,  3,  0;
  4,  6,  6,  4;
  3,  1,  5,  1,   3;
  2,  4,  1,  1,   4,  2;
  1,  8,  8, 11,   8,  8,   1;
  0,  0, 16, 18,  18, 16,   0,   0;
  6,  4,  1, 15,   4, 15,   1,   4,  6;
  5,  9,  4, 31,  22, 22,  31,   4,  9,  5;
  4, 15,  5, 34,  49, 37,  49,  34,  5, 15,   4;
  3,  3,  3, 14,   9, 48,  48,   9, 14,  3,   3,  3;
  2,  9, 36, 23,  23, 22,  49,  22, 23, 23,  36,  9,  2;
  1, 16, 29,  4,  22, 36, 126, 126, 36, 22,   4, 29, 16, 1;
  0,  1, 16, 29, 121, 92,   9, 126,  9, 92, 121, 29, 16, 1, 0;
		

Links

• Seiichi Manyama, Rows n = 2..141, flattened
• Eric Weisstein's World of Mathematics, Binomial Coefficient

Crossrefs

Column k=1..2 give A068527, A175032.

Cf. A001108, A007318.

Programs

Mathematica

diff[n_] := Ceiling[Sqrt[n]]^2 - n; T[n_, k_] := diff @ Binomial[n, k]; Table[T[n, k], {n, 2, 14}, {k, 1, n - 1}] // Flatten (* Amiram Eldar, Apr 03 2021 *)

PARI

T(n, k) = my(m=binomial(n, k)); if(issquare(m), 0, (sqrtint(m)+1)^2-m);

Formula

T(n,k) = T(n,n-k) = A068527(binomial(n,k)).

T(n^2,1) = T(n^2,n^2-1) = 0.

If 3 <= k <= n-3 and (n,k) is not (50,3) or (50,47), T(n,k) > 0.