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.

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, 49, 32, 17, 4, 7, 16, 23, 28, 64, 45, 28, 13, 0, 11, 20, 27, 32, 81, 60, 41, 24, 9, 4, 15, 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

Offset: 0

Author

Stefano Spezia, Mar 21 2025

Keywords

• nonn
• easy
• tabl

Examples

			The array begins as:
   0,  4,  8, 12, 16, 20, 24, 28, 32, 36, 40, 44, ...
   1,  3,  7, 11, 15, 19, 23, 27, 31, 35, 39, 43, ...
   4,  0,  4,  8, 12, 16, 20, 24, 28, 32, 36, 40, ...
   9,  5,  1,  3,  7, 11, 15, 19, 23, 27, 31, 35, ...
  16, 12,  8,  4,  0,  4,  8, 12, 16, 20, 24, 28, ...
  25, 21, 17, 13,  9,  5,  1,  3,  7, 11, 15, 19, ...
  ...
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;
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.
		

Crossrefs

Cf. A000290 (m=0), A008586 (n=0), A028347, A028566, A028884, A131098, A134594, A145917, A382311 (antidiagonal sums).

Programs

Mathematica

A[n_,m_]:=Abs[n^2-4m]; Table[A[n-m,m],{n,0,11},{m,0,n}]//Flatten

Formula

A(n,m) = abs(n^2 - 4*m).

A(n,n) = A028347(n-2) for n > 3.

A(n,1) = A028347(n) for n > 1.

A(n,2) = A028884(n-3) for n > 2.

A(n,4) = A028566(n-4) for n > 3.

A(n,5) = A134594(n-5) for n > 4.

A(1,n) = A131098(n+1).