A055088 - OEIS

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

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OFFSET

1,1

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".

LINKS

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

EXAMPLE

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]

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 *)

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

CROSSREFS

Cf. A002144, A002145, A011582, A011583.

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