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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).

%I A343037 #37 Apr 04 2021 12:39:17
%S A343037 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,
%T A343037 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,
%U A343037 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
%N 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).
%H A343037 Seiichi Manyama, <a href="/A343037/b343037.txt">Rows n = 2..141, flattened</a>
%H A343037 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialCoefficient.html">Binomial Coefficient</a>
%F A343037 T(n,k) = T(n,n-k) = A068527(binomial(n,k)).
%F A343037 T(n^2,1) = T(n^2,n^2-1) = 0.
%F A343037 If 3 <= k <= n-3 and (n,k) is not (50,3) or (50,47), T(n,k) > 0.
%e A343037 binomial(50,3) = binomial(50,47) = 140^2. So T(50,3) = T(50,47) = 0.
%e A343037 Triangle begins:
%e A343037   2;
%e A343037   1,  1;
%e A343037   0,  3,  0;
%e A343037   4,  6,  6,  4;
%e A343037   3,  1,  5,  1,   3;
%e A343037   2,  4,  1,  1,   4,  2;
%e A343037   1,  8,  8, 11,   8,  8,   1;
%e A343037   0,  0, 16, 18,  18, 16,   0,   0;
%e A343037   6,  4,  1, 15,   4, 15,   1,   4,  6;
%e A343037   5,  9,  4, 31,  22, 22,  31,   4,  9,  5;
%e A343037   4, 15,  5, 34,  49, 37,  49,  34,  5, 15,   4;
%e A343037   3,  3,  3, 14,   9, 48,  48,   9, 14,  3,   3,  3;
%e A343037   2,  9, 36, 23,  23, 22,  49,  22, 23, 23,  36,  9,  2;
%e A343037   1, 16, 29,  4,  22, 36, 126, 126, 36, 22,   4, 29, 16, 1;
%e A343037   0,  1, 16, 29, 121, 92,   9, 126,  9, 92, 121, 29, 16, 1, 0;
%t A343037 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 *)
%o A343037 (PARI) T(n, k) = my(m=binomial(n, k)); if(issquare(m), 0, (sqrtint(m)+1)^2-m);
%Y A343037 Column k=1..2 give A068527, A175032.
%Y A343037 Cf. A001108, A007318.
%K A343037 nonn,tabl
%O A343037 2,1
%A A343037 _Seiichi Manyama_, Apr 03 2021