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A382310 Array read by ascending antidiagonals: A(n,m) is the squared distance between the roots of the 2nd degree equations z^2 +- n*z + m = 0 on the complex plane.

%I A382310 #16 Mar 22 2025 19:22:03
%S A382310 0,1,4,4,3,8,9,0,7,12,16,5,4,11,16,25,12,1,8,15,20,36,21,8,3,12,19,24,
%T A382310 49,32,17,4,7,16,23,28,64,45,28,13,0,11,20,27,32,81,60,41,24,9,4,15,
%U A382310 24,31,36,100,77,56,37,20,5,8,19,28,35,40,121,96,73,52,33,16,1,12,23,32,39,44
%N A382310 Array read by ascending antidiagonals: A(n,m) is the squared distance between the roots of the 2nd degree equations z^2 +- n*z + m = 0 on the complex plane.
%F A382310 A(n,m) = abs(n^2 - 4*m).
%F A382310 A(n,n) = A028347(n-2) for n > 3.
%F A382310 A(n,1) = A028347(n) for n > 1.
%F A382310 A(n,2) = A028884(n-3) for n > 2.
%F A382310 A(n,4) = A028566(n-4) for n > 3.
%F A382310 A(n,5) = A134594(n-5) for n > 4.
%F A382310 A(1,n) = A131098(n+1).
%e A382310 The array begins as:
%e A382310    0,  4,  8, 12, 16, 20, 24, 28, 32, 36, 40, 44, ...
%e A382310    1,  3,  7, 11, 15, 19, 23, 27, 31, 35, 39, 43, ...
%e A382310    4,  0,  4,  8, 12, 16, 20, 24, 28, 32, 36, 40, ...
%e A382310    9,  5,  1,  3,  7, 11, 15, 19, 23, 27, 31, 35, ...
%e A382310   16, 12,  8,  4,  0,  4,  8, 12, 16, 20, 24, 28, ...
%e A382310   25, 21, 17, 13,  9,  5,  1,  3,  7, 11, 15, 19, ...
%e A382310   ...
%e A382310 A(2,0) = 4 since z^2 - 2*z = 0 and z^2 + 2*z = 0 have respectively roots 0, 2, and -2, 0 with squared distance equal to 4;
%e A382310 A(1,2) = 7 since z^2 - z + 2 = 0 and z^2 + z + 2 = 0 have respectively roots (1 +- i*sqrt(7))/2 and (-1 +- i*sqrt(7))/2 with squared distance equal to 7, where i denotes the imaginary unit.
%t A382310 A[n_,m_]:=Abs[n^2-4m]; Table[A[n-m,m],{n,0,11},{m,0,n}]//Flatten
%Y A382310 Cf. A000290 (m=0), A008586 (n=0), A028347, A028566, A028884, A131098, A134594, A145917, A382311 (antidiagonal sums).
%K A382310 nonn,easy,tabl
%O A382310 0,3
%A A382310 _Stefano Spezia_, Mar 21 2025