A385104 Triangle read by rows: T(n,k) is the number of residue classes obtained by solving mod(x^2,n) = k for x over the integers, n >= 1, k >= 0.

1, 1, 1, 1, 2, 0, 2, 2, 0, 0, 1, 2, 0, 0, 2, 1, 2, 0, 1, 2, 0, 1, 2, 2, 0, 2, 0, 0, 2, 4, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 2, 0, 1, 2, 0, 0, 2, 1, 2, 0, 0, 2, 1, 2, 0, 2, 2, 2, 0, 0, 0, 2, 0, 2, 4, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 1, 2, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 2, 0, 2, 0, 0, 1, 2

Offset: 1

Author

Jason Bard, Jun 18 2025

Keywords

• nonn
• tabl

Comments

The sum of each row is n.

Examples

			Triangle starts:
  1
  1 1
  1 2 0
  2 2 0 0
  1 2 0 0 2
  1 2 0 1 2 0
  1 2 2 0 2 0 0
  2 4 0 0 2 0 0 0
  3 2 0 0 2 0 0 2 0
  1 2 0 0 2 1 2 0 0 2
  ...
		

Links

• Jason Bard, Table of n, a(n) for n = 1..11325

Crossrefs

Cf. A000089, A000188, A060594, A096008.

Programs

Mathematica

dat[n_] := Table[Reduce[Mod[x^2, n] == k, x, Integers], {k, 0, n - 1}]; countConditions[cond_] := Which[cond === False, 0, MatchQ[cond, x \[Element] Integers], 1, True, Length@Cases[cond, Equal[x, _], Infinity]]; counts = Flatten[Table[countConditions /@ dat[n], {n, 1, 20}]]

PARI

T(n, k) = sum(i=1, n, Mod(i,n)^2 == k);
row(n) = vector(n, i, T(n, i-1)); \\ Michel Marcus, Jun 23 2025

Formula

T(n,0) = A000188(n).

T(n,1) = A060594(n).

T(n,n-1) = A000089(n).