This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382252 #9 Apr 16 2025 10:25:15
%S A382252 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,
%T A382252 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,
%U A382252 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
%N A382252 Triangle T(n,k) = numerator of (n+k)/(1+n*k), 0 <= k <= n >= 0, read by rows.
%C A382252 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.
%F A382252 T(n,k) = T(k,n) for all n, k >= 0; therefore only k <= n is considered here.
%F A382252 T(n,0) = T(0,n) = n and T(n,1) = T(1,n) = 1 for all n >= 0.
%F A382252 T(n,n) = A022998(n) = n if odd, else 2*n.
%e A382252 The table for the operation n @ k := (n + k)/(1 + n*k) starts as follows:
%e A382252 (0 is the neutral element for the operation: n @ 0 = n = 0 @ n, therefore row and column 0 give the column and row headers.)
%e A382252 0 1 2 3 4 5 6 7 8 Numerators of 0;
%e A382252 1 1 1 1 1 1 1 1 1 lower left 1, 1:
%e A382252 2 1 4/5 5/7 2/3 7/11 8/13 3/5 10/17 triangle: 2, 1, 4;
%e A382252 3 1 5/7 3/5 7/13 1/2 9/19 5/11 11/25 3, 1, 5, 3
%e A382252 4 1 2/3 7/13 8/17 3/7 2/5 11/29 4/11 4, 1, 2, 7, 8;
%e A382252 5 1 7/11 1/2 3/7 5/13 11/31 1/3 13/41 etc.
%e A382252 6 1 8/13 9/19 2/5 11/31 12/37 13/43 2/7
%e A382252 7 1 3/5 5/11 11/29 1/3 13/43 7/25 5/19
%e A382252 8 1 10/17 11/25 4/11 13/41 2/7 5/19 16/65
%e A382252 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.
%o A382252 (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])
%Y A382252 Cf. A382253 (denominators), A382257 (related); A228564 (main diagonal), A001477 (row & col. 0), A000012 (row & col. 1).
%K A382252 nonn,tabl,frac
%O A382252 0,4
%A A382252 _M. F. Hasler_, Apr 15 2025