A055088 Triangle of generalized Legendre symbols L(a/b) read by rows, with 1's for quadratic residues and 0's for quadratic non-residues.

1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0

Offset: 1

Author

Antti Karttunen, Apr 18 2000

Keywords

• nonn
• tabl

Comments

L(a/b) is 1 if an integer c exists such that c^2 is congruent to a (mod b) and 0 otherwise.

For every prime of the form 4k+1 (A002144) the row is symmetric and for every prime of the form 4k+3 (A002145) the row is "complementarily symmetric".

Examples

			The tenth row gives the quadratic residues and non-residues of 11 (see A011582) and the twelfth row gives the same information for 13 (A011583), with -1's replaced by zeros.
.
Triangle starts:
  [ 1] [1]
  [ 2] [1, 0]
  [ 3] [1, 0, 0]
  [ 4] [1, 0, 0, 1]
  [ 5] [1, 0, 1, 1, 0]
  [ 6] [1, 1, 0, 1, 0, 0]
  [ 7] [1, 0, 0, 1, 0, 0, 0]
  [ 8] [1, 0, 0, 1, 0, 0, 1, 0]
  [ 9] [1, 0, 0, 1, 1, 1, 0, 0, 1]
  [10] [1, 0, 1, 1, 1, 0, 0, 0, 1, 0]
  [11] [1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0]
  [12] [1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1]
		

Crossrefs

Cf. A002144, A002145, A011582, A011583.

Each row interpreted as a binary number: A055094.

Programs

Maple

# See A054431 for one_or_zero and trinv.
with(numtheory,quadres); quadres_0_1_array := (n) -> one_or_zero(quadres((n-((trinv(n-1)*(trinv(n-1)-1))/2)), (trinv(n-1)+1)));

Mathematica

row[n_] := With[{rr = Table[Mod[k^2, n + 1], {k, 1, n}] // Union}, Boole[ MemberQ[rr, #]]& /@ Range[n]];
Array[row, 14] // Flatten (* Jean-François Alcover, Mar 05 2016 *)

Sage

def A055088_row(n) :
    Q = quadratic_residues(n+1)
    return [int(i in Q) for i in (1..n)]
for n in (1..14) : print(A055088_row(n))  # Peter Luschny, Aug 08 2012