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A055088
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Triangle of generalized Legendre symbols L(a/b), with 1's for quadratic residues and 0's for quadratic non-residues.
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4
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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
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1,1
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COMMENTS
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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".
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LINKS
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Table of n, a(n) for n=1..105.
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FORMULA
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[seq(quadres_0_1_array(j), j=1..)]; (see Maple code below)
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EXAMPLE
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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.
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MAPLE
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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)));
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PROG
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(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
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CROSSREFS
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Cf. A054431 for one_or_zero and trinv. Each row interpreted as a binary number: A055094.
Sequence in context: A033788 A033782 A033778 * A068427 A176918 A176890
Adjacent sequences: A055085 A055086 A055087 * A055089 A055090 A055091
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KEYWORD
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nonn,tabl
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AUTHOR
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Antti Karttunen Apr 18 2000
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STATUS
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approved
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