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