A382252 Triangle T(n,k) = numerator of (n+k)/(1+n*k), 0 <= k <= n >= 0, read by rows.
0, 1, 1, 2, 1, 4, 3, 1, 5, 3, 4, 1, 2, 7, 8, 5, 1, 7, 1, 3, 5, 6, 1, 8, 9, 2, 11, 12, 7, 1, 3, 5, 11, 1, 13, 7, 8, 1, 10, 11, 4, 13, 2, 5, 16, 9, 1, 11, 3, 13, 7, 3, 1, 17, 9, 10, 1, 4, 13, 14, 5, 16, 17, 2, 19, 20, 11, 1, 13, 7, 1, 2, 17, 3, 19, 1, 7, 11, 12, 1, 14, 15, 16, 17, 18, 19, 20, 21, 2, 23, 24
Offset: 0
Examples
The table for the operation n @ k := (n + k)/(1 + n*k) starts as follows: (0 is the neutral element for the operation: n @ 0 = n = 0 @ n, therefore row and column 0 give the column and row headers.) 0 1 2 3 4 5 6 7 8 Numerators of 0; 1 1 1 1 1 1 1 1 1 lower left 1, 1: 2 1 4/5 5/7 2/3 7/11 8/13 3/5 10/17 triangle: 2, 1, 4; 3 1 5/7 3/5 7/13 1/2 9/19 5/11 11/25 3, 1, 5, 3 4 1 2/3 7/13 8/17 3/7 2/5 11/29 4/11 4, 1, 2, 7, 8; 5 1 7/11 1/2 3/7 5/13 11/31 1/3 13/41 etc. 6 1 8/13 9/19 2/5 11/31 12/37 13/43 2/7 7 1 3/5 5/11 11/29 1/3 13/43 7/25 5/19 8 1 10/17 11/25 4/11 13/41 2/7 5/19 16/65 This sequence lists the numerators of the values, where numerator(x) = x for integers, and only for the lower left triangle of the table, by rows.
Crossrefs
Programs
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PARI
apply( {A382252(n,k=-1)= k<0&& k=n-(1+n=(sqrtint(8*n+1)-1)\2)*n/2; numerator((n+k)/(1+n*k))}, [0..30])
Formula
T(n,k) = T(k,n) for all n, k >= 0; therefore only k <= n is considered here.
T(n,0) = T(0,n) = n and T(n,1) = T(1,n) = 1 for all n >= 0.
T(n,n) = A022998(n) = n if odd, else 2*n.
Comments