A373354 Triangle read by rows: T(n, k) = [n - k + 1 | k] where [n | k] is defined below.

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

Offset

1,8

Links

Table of n, a(n) for n=1..91.

Carl Friedrich Gauss, Vierter Abschnitt. Von den Congruenzen zweiten Grades. Quadratische Reste und Nichtreste. Art. 97, in "Untersuchungen über die höhere Arithmetik", Hrsg. H. Maser, Verlag von Julius Springer, Berlin, 1889.

Formula

Let two positive numbers n, k be given. We write (n R k) if two integers x and y exist, such that x^2 = n + k*y, and (n N k) otherwise. If the condition is satisfied n is called a quadratic residue modulo k. We distinguish four cases:

[n | k] := 0 if (n N k) and (k N n);

[n | k] := 1 if (n R k) and (k R n);

[n | k] := 2 if (n R k) and (k N n);

[n | k] := 3 if (n N k) and (k R n).

We set T(n, k) = [n - k + 1 | k].

Exchanging 2 <-> 3 reverses the rows.

All terms of row n are 1 <==> n = 1, 2 or n is of the form k*(k-2), k >= 3.

Example

Triangle starts:
  [ 1] 1;
  [ 2] 1, 1;
  [ 3] 1, 1, 1;
  [ 4] 1, 2, 3, 1;
  [ 5] 1, 2, 1, 3, 1;
  [ 6] 1, 2, 2, 3, 3, 1;
  [ 7] 1, 2, 0, 1, 0, 3, 1;
  [ 8] 1, 1, 1, 1, 1, 1, 1, 1;
  [ 9] 1, 2, 2, 3, 1, 2, 3, 3, 1;
  [10] 1, 2, 0, 3, 2, 3, 2, 0, 3, 1;
  [11] 1, 2, 2, 1, 0, 1, 0, 1, 3, 3, 1;
  [12] 1, 2, 2, 1, 0, 2, 3, 0, 1, 3, 3, 1;

Maple

QRS := proc(n, k) local QR, p, q, a, b;
   QR := (a, n) -> NumberTheory:-QuadraticResidue(a, n);
   a := QR(n, k); b := QR(k, n);
   if a = -1 and b = -1 then return 0 fi;
   if a =  1 and b =  1 then return 1 fi;
   if a =  1 and b = -1 then return 2 fi;
   if a = -1 and b =  1 then return 3 fi;
end: for n from 1 to 12 do lprint([n], seq(QRS(n-k+1, k), k = 1..n)) od;

Prog

(Python)
from sympy.ntheory import is_quad_residue
def QR(n, k): return is_quad_residue(n, k)
def QRS(n, k):
    a = QR(n, k); b = QR(k, n)
    if not a and not b: return 0
    if a and b: return 1
    if a and not b: return 2
    if not a and b: return 3
def T(n, k): return QRS(n - k + 1, k)
for n in range(1, 13): print([n], [T(n, k) for k in range(1, n + 1)])

Crossrefs

Cf. A373223, A373355 (restricted to primes).

Sequence in context: A372571 A191770 A120966 * A189041 A055444 A102383

Adjacent sequences: A373351 A373352 A373353 * A373355 A373356 A373357

Keyword

nonn,tabl

Author

Peter Luschny, Jun 02 2024

Status

approved