A233558 Triangle read by rows: T(n,k) = real part mod n of (n + ki)^2, where k=1..n-1 and i is the imaginary unit.

1, 2, 2, 3, 0, 3, 4, 1, 1, 4, 5, 2, 3, 2, 5, 6, 3, 5, 5, 3, 6, 7, 4, 7, 0, 7, 4, 7, 8, 5, 0, 2, 2, 0, 5, 8, 9, 6, 1, 4, 5, 4, 1, 6, 9, 10, 7, 2, 6, 8, 8, 6, 2, 7, 10, 11, 8, 3, 8, 11, 0, 11, 8, 3, 8, 11, 12, 9, 4, 10, 1, 3, 3, 1, 10, 4, 9, 12, 13, 10, 5, 12, 3, 6, 7, 6, 3, 12, 5, 10, 13, 14, 11, 6, 14, 5, 9, 11, 11, 9, 5, 14

Offset

2,2

Comments

For prime n, if n == 1 (mod 4), sequence gives quadratic residues of n, and if n == 3 (mod 4) the sequence gives non-quadratic residues.

Check: The sixth row, the row for 7 (of the form 4k+3): {6, 3, 5, 5, 3, 6} contains only quadratic non-residues (3, 5 and 6).

Also, on the tenth row, for 11 (of the form 4k+3 also) it is also true: {10, 7, 2, 6, 8, 8, 6, 2, 7, 10}, as 2, 6, 7, 8 and 10 are exactly the quadratic non-residues of 11.

Also, on the twelfth row, for n=13 (of the form 4k+1), it is true that all its quadratic residues are listed: {12, 9, 4, 10, 1, 3, 3, 1, 10, 4, 9, 12}.

Links

Table of n, a(n) for n=2..103.

Formula

As a table array with offset 1, T(n, k) = (n*k) % (n+k). - Michel Marcus, Nov 28 2019

Example

Triangle starts:
  1;
  2, 2;
  3, 0, 3;
  4, 1, 1, 4;
  5, 2, 3, 2, 5;
  6, 3, 5, 5, 3, 6;

Prog

(JavaScript)
function cNumber(x, y) {
return [x, y];
}
function cMult(a, b) {
return [a[0]*b[0]-a[1]*b[1], a[0]*b[1]+a[1]*b[0]];
}
for (i=1; i<20; i++)
for (j=1; j<i; j++) {
ij=cNumber(i, j);
ij2=cMult(ij, ij);
ijm=ij2[0]%ij[0];
document.write(ijm+", ");
}
(PARI) T(n, k) = real((n+k*I)^2) % n;
tabl(nn) = for(n=2, nn, for(k=1, n-1, print1(T(n, k), ", "); ); print(); ) \\ Michel Marcus, Jun 25 2018

Crossrefs

Sequence in context: A297114 A318448 A103516 * A319929 A129234 A213081

Adjacent sequences: A233555 A233556 A233557 * A233559 A233560 A233561

Keyword

nonn,tabl

Author

Jon Perry, Dec 13 2013

Extensions

Edited by Jon E. Schoenfield, Jun 30 2018

Status

approved