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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.)
4
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.
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jun 23 2024
STATUS
approved