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A226520
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Irregular triangle read by rows: T(n,k) = Legendre(k,prime(n)), for 0 <= k < prime(n).
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7
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0, 1, 0, 1, -1, 0, 1, -1, -1, 1, 0, 1, 1, -1, 1, -1, -1, 0, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 0, 1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, 0, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 0, 1, -1, -1, 1, 1, 1, 1, -1, 1, -1, 1, -1, -1, -1, -1, 1, 1, -1
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OFFSET
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1
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COMMENTS
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Strictly speaking, the symbol in the definition is the Legendre-Jacobi-Kronecker symbol, since the Legendre symbol is defined only for odd primes.
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REFERENCES
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R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; p. 289.
Beck, József. Inevitable randomness in discrete mathematics. University Lecture Series, 49. American Mathematical Society, Providence, RI, 2009. xii+250 pp. ISBN: 978-0-8218-4756-5; MR2543141 (2010m:60026). See page 23.
H. Cohen, A Course in Computational Alg. No. Theory, Springer, 1993, p. 28.
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins
[0, 1],
[0, 1, -1],
[0, 1, -1, -1, 1],
[0, 1, 1, -1, 1, -1, -1],
[0, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1],
[0, 1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1],
[0, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1],
...
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MAPLE
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with(numtheory);
T:=(n, k)->legendre(k, ithprime(n));
f:=n->[seq(T(n, k), k=0..ithprime(n)-1)];
[seq(f(n), n=1..15)];
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MATHEMATICA
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Table[p = Prime[n]; Table[JacobiSymbol[k, p], {k, 0, p-1}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Mar 07 2014 *)
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PROG
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(Haskell)
a226520 n k = a226520_tabf !! (n-1) !! k
a226520_row n = a226520_tabf !! (n-1)
a226520_tabf =
map (\p -> map (flip legendreSymbol p) [0..p-1]) a000040_list
-- where the function legendreSymbol is defined in A097343.
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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