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

2,1

Links

Seiichi Manyama, Rows n = 2..141, flattened

Eric Weisstein's World of Mathematics, Binomial Coefficient

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.

Example

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;

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 *)

Prog

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

Crossrefs

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

Cf. A001108, A007318.

Sequence in context: A257991 A373592 A343029 * A152434 A143810 A128589

Adjacent sequences: A343034 A343035 A343036 * A343038 A343039 A343040

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 03 2021

Status

approved