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A372877
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Jacobi's triangle read by rows. T(n, k) = J(n, k) where J(n, k) is the Jacobi symbol (n / k) provided k is odd and otherwise 0.
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6
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0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, -1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0
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OFFSET
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0
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COMMENTS
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Normally J(n, k) is only defined for odd k. Here, J(n, k) is 0 if and only if k is not prime to n or k is even. If k is an odd prime, J(n, k) is equal to the Legendre symbol L(n, k). If k is odd, J(n, k) is equal to the Kronecker symbol K(n, k). If n is a square residue mod k then J(n, k) = 1; but conversely it does not follow from J(n, k) = 1 that n is a square residue mod k.
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LINKS
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Carl Gustav Jacob Jacobi, Über die Kreistheilung und ihre Anwendung auf die Zahlentheorie, Bericht Akad. d. Wiss. zu Berlin 1837. Reprinted in Journal für die reine und angewandte Mathematik 30 (1846): 166-182.
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EXAMPLE
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Triangle J(n, k) starts:
[0] 0;
[1] 0, 1;
[2] 0, 1, 0;
[3] 0, 1, 0, 0;
[4] 0, 1, 0, 1, 0;
[5] 0, 1, 0, -1, 0, 0;
[6] 0, 1, 0, 0, 0, 1, 0;
[7] 0, 1, 0, 1, 0, -1, 0, 0;
[8] 0, 1, 0, -1, 0, -1, 0, 1, 0;
[9] 0, 1, 0, 0, 0, 1, 0, 1, 0, 0;
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Not limiting the range of k leads to the square array:
[0] 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
[1] 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...
[2] 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, ...
[3] 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, ...
[4] 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...
[5] 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, ...
[6] 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, ...
[7] 0, 1, 0, 1, 0, -1, 0, 0, 0, 1, ...
[8] 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, ...
[9] 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, ...
...
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MAPLE
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J := (n, k) -> ifelse(k::even, 0, NumberTheory:-JacobiSymbol(n, k)):
seq(seq(J(n, k), k = 0..n), n = 0..12);
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MATHEMATICA
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Table[If[EvenQ[k], 0, JacobiSymbol[n, k]], {n, 0, 12}, {k, 0, n}] // Flatten
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PROG
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(Python) # This Jacobi symbol is also defined for even k and n = 0.
def JacobiSymbol(n, k):
if n == 0 and k == 1: return 1
if n == 0 or k % 2 == 0: return 0
n %= k
res = 1
while n != 0:
while n % 2 == 0:
n //= 2
if k % 8 in (3, 5): res = -res
if n % 4 == 3 == k % 4: res = -res
n, k = k % n, n
return res if k == 1 else 0
for n in range(10): print([JacobiSymbol(n, k) for k in range(n + 1)])
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CROSSREFS
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Family: A217831 (Euclid's triangle), A372728 (Kronecker's triangle), A372726 (Legendre's triangle), A373223 (Gauss' triangle), A373751 (quadratic residue modulo prime(n)), A373748 (quadratic residue/nonresidue modulo n).
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KEYWORD
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AUTHOR
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STATUS
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approved
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