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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). 1
 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 (list; table; graph; refs; listen; history; text; internal format)
 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: A318808 A257991 A343029 * A152434 A143810 A128589 Adjacent sequences:  A343034 A343035 A343036 * A343038 A343039 A343040 KEYWORD nonn,tabl AUTHOR Seiichi Manyama, Apr 03 2021 STATUS approved

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Last modified August 2 20:38 EDT 2021. Contains 346428 sequences. (Running on oeis4.)