A373749 Triangle read by rows: T(n, k) = MOD(k^2, n) where MOD(a, n) = a if n = 0 and otherwise a - n*floor(a/n). (Quadratic residue.)

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 4, 3, 4, 1, 0, 0, 1, 4, 2, 2, 4, 1, 0, 0, 1, 4, 1, 0, 1, 4, 1, 0, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 0, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1, 0, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0

Offset

0,18

Comments

The definition of the binary operation MOD in the name follows CMath (Graham et al.) and Bach & Shallit. This is important because some CAS unfortunately do not follow this definition and throw a 'division by zero' error if n = 0.

Row n reduced to a set is the set of the quadratic residues mod n.

References

Eric Bach and Jeffrey Shallit, Algorithmic Number Theory, p. 21.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., pp. 81f.

Links

Table of n, a(n) for n=0..90.

Example

Triangle starts:
  [0] 0;
  [1] 0, 0;
  [2] 0, 1, 0;
  [3] 0, 1, 1, 0;
  [4] 0, 1, 0, 1, 0;
  [5] 0, 1, 4, 4, 1, 0;
  [6] 0, 1, 4, 3, 4, 1, 0;
  [7] 0, 1, 4, 2, 2, 4, 1, 0;
  [8] 0, 1, 4, 1, 0, 1, 4, 1, 0;
  [9] 0, 1, 4, 0, 7, 7, 0, 4, 1, 0;
 [10] 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0;

Maple

REM := (n, k) -> ifelse(k = 0, n, irem(n, k)):
T := n -> local k; seq(REM(k^2, n), k = 0..n):
seq(T(n), n = 0..12);

Mathematica

MOD[n_, k_] := If[k == 0, n, Mod[n, k]];
Table[MOD[k^2, n], {n, 0, 10}, {k, 0, n}]

Prog

(Julia)
Mod(n, k) = k == 0 ? n : mod(n, k)
T(n, k) = Mod(k^2, n)
for n in 0:10
    [T(n, k) for k in 0:n] |> println
end
(SageMath)
def A373749(n, k): return mod(k^2, n)
for n in range(11): print([A373749(n, k) for k in range(n + 1)])

Crossrefs

Variants: A048152, A096008.

Cf. A048153 (row sums), A373750 (middle terms).

Sequence in context: A260043 A366467 A366464 * A185057 A343720 A048152

Adjacent sequences: A373746 A373747 A373748 * A373750 A373751 A373752

Keyword

nonn,tabl

Author

Peter Luschny, Jun 23 2024

Status

approved