OFFSET
2,1
COMMENTS
Row sums = 0. (p,k)==k^((p-1)/2) (mod p). For example, row n=4 of the triangle (for the 4th prime p = 7) reads: 1,1,-1,1,-1,-1,0 because 1^3==1, 2^3==1, 3^3==-1, 4^3==1, 5^3==-1, 6^3==-1, 7^3==0 (mod 7). - Geoffrey Critzer, Apr 18 2015
LINKS
Reinhard Zumkeller, Rows n = 2..75 of triangle, flattened
Haskell for Math, Number Theory Fundamentals
Wikipedia, Legendre symbol
FORMULA
(p, p)=0, all others are +- 1.
EXAMPLE
MAPLE
with(numtheory):
T:= n-> (p-> seq(jacobi(k, p), k=1..p))(ithprime(n)):
seq(T(n), n=2..15); # Alois P. Heinz, Apr 19 2015
MATHEMATICA
Flatten[ Table[ JacobiSymbol[ Range[ Prime[n]], Prime[n]], {n, 2, 8}]]
PROG
(Haskell)
a097343 n k = a097343_tabf !! (n-2) !! (k-1)
a097343_row n = a097343_tabf !! (n-2)
a097343_tabf =
map (\p -> map (flip legendreSymbol p) [1..p]) $ tail a000040_list
legendreSymbol a p = if a' == 0 then 0 else twoSymbol * oddSymbol where
a' = a `mod` p
(s, q) = a' `splitWith` 2
twoSymbol = if (p `mod` 8) `elem` [1, 7] || even s then 1 else -1
oddSymbol = if q == 1 then 1 else qrMultiplier * legendreSymbol p q
qrMultiplier = if p `mod` 4 == 3 && q `mod` 4 == 3 then -1 else 1
splitWith n p = spw 0 n where
spw s t = if m > 0 then (s, t) else spw (s + 1) t'
where (t', m) = divMod t p
-- See link. Reinhard Zumkeller, Feb 02 2014
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Robert G. Wilson v, Aug 02 2004
STATUS
approved