OFFSET
0,4
COMMENTS
Since the operation n @ k := (n + k)/(1 + n*k) is commutative, it is sufficient to list only the lower half of the "multiplication table", which would otherwise be an infinite square array. This triangle lists the numerators, and A382253 lists the denominators.
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.
EXAMPLE
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.
PROG
(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])
CROSSREFS
KEYWORD
AUTHOR
M. F. Hasler, Apr 15 2025
STATUS
approved