A055088 Triangle of generalized Legendre symbols L(a/b), 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

1,1

Comments

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".

Links

Table of n, a(n) for n=1..105.

Formula

[seq(quadres_0_1_array(j), j=1..)]; (see Maple code below)

Example

Terms are L(1/2); L(1/3), L(2/3); L(1/4), L(2/4), L(3/4); L(1/5), ... where L(a/b) is 1 if an integer c exists such that c^2 is congruent to a (mod b) and 0 otherwise.
E.g. 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.

Maple

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 *)

Prog

(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) : A055088_row(n)  # Peter Luschny, Aug 08 2012

Crossrefs

Cf. A054431 for one_or_zero and trinv. Each row interpreted as a binary number: A055094.

Sequence in context: A033788 A329680 A257234 * A266666 A068427 A190191

Adjacent sequences: A055085 A055086 A055087 * A055089 A055090 A055091

Keyword

nonn,tabl

Author

Antti Karttunen, Apr 18 2000

Status

approved