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A354522 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = g(f(n) + f(k)) where f denotes A001057 and g denotes its inverse.

%I A354522 #34 Sep 18 2022 12:37:54
%S A354522 0,1,1,2,3,2,3,0,0,3,4,5,4,5,4,5,2,1,1,2,5,6,7,6,7,6,7,6,7,4,3,0,0,3,
%T A354522 4,7,8,9,8,9,8,9,8,9,8,9,6,5,2,1,1,2,5,6,9,10,11,10,11,10,11,10,11,10,
%U A354522 11,10,11,8,7,4,3,0,0,3,4,7,8,11,12,13,12,13,12,13,12,13,12,13,12,13,12
%N A354522 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = g(f(n) + f(k)) where f denotes A001057 and g denotes its inverse.
%C A354522 This sequence is directly related to A355278.
%C A354522 The function f is a bijection from the nonnegative integers to the integers (Z).
%C A354522 The nonnegative integers, together with (x,y) -> A(x,y), form an abelian group isomorph to the additive group Z (f and g act as isomorphisms).
%C A354522 As a consequence, each row and each column is a permutation of the nonnegative integers.
%F A354522 A355278(n+1, k+1) = prime(1 + A(n, k)) (where prime(m) denotes the m-th prime number).
%F A354522 A(n, k) = A(k, n).
%F A354522 A(n, 0) = n.
%F A354522 A(n, A014681(n)) = 0.
%F A354522 A(m, A(n, k)) = A(A(m, n), k).
%F A354522 A(n, n) = A014601(n).
%F A354522 A(n, A(n, n)) = A047264(n+1).
%F A354522 A(A(n, n), A(n, n)) = A047521(n+1).
%e A354522 Array A(n, k) begins:
%e A354522   n\k |  0   1   2   3   4   5   6   7   8   9  10  11  12
%e A354522   ----+---------------------------------------------------
%e A354522     0 |  0   1   2   3   4   5   6   7   8   9  10  11  12
%e A354522     1 |  1   3   0   5   2   7   4   9   6  11   8  13  10
%e A354522     2 |  2   0   4   1   6   3   8   5  10   7  12   9  14
%e A354522     3 |  3   5   1   7   0   9   2  11   4  13   6  15   8
%e A354522     4 |  4   2   6   0   8   1  10   3  12   5  14   7  16
%e A354522     5 |  5   7   3   9   1  11   0  13   2  15   4  17   6
%e A354522     6 |  6   4   8   2  10   0  12   1  14   3  16   5  18
%e A354522     7 |  7   9   5  11   3  13   1  15   0  17   2  19   4
%e A354522     8 |  8   6  10   4  12   2  14   0  16   1  18   3  20
%e A354522     9 |  9  11   7  13   5  15   3  17   1  19   0  21   2
%e A354522    10 | 10   8  12   6  14   4  16   2  18   0  20   1  22
%e A354522    11 | 11  13   9  15   7  17   5  19   3  21   1  23   0
%e A354522    12 | 12  10  14   8  16   6  18   4  20   2  22   0  24
%o A354522 (PARI) f(n) = - (-1)^n * ((n+1)\2)
%o A354522 g(n) = if (n<=0, -2*n, 2*n-1)
%o A354522 A(n, k) = g(f(n) + f(k))
%Y A354522 Cf. A001057, A014601, A014681, A047264, A047521, A355278, A357144.
%K A354522 nonn,tabl
%O A354522 0,4
%A A354522 _Rémy Sigrist_, Sep 14 2022