A372877 - OEIS

A372877

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.

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 (list; table; graph; refs; listen; history; text; internal format)

	OFFSET	`0`
	COMMENTS	`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.`
	LINKS	`Table of n, a(n) for n=0..90.` `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.`
	EXAMPLE	`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;` `.` `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, ...` `...`
	MAPLE	`J := (n, k) -> ifelse(k::even, 0, NumberTheory:-JacobiSymbol(n, k)):` `seq(seq(J(n, k), k = 0..n), n = 0..12);`
	MATHEMATICA	`Table[If[EvenQ[k], 0, JacobiSymbol[n, k]], {n, 0, 12}, {k, 0, n}] // Flatten`
	PROG	`(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)])`
	CROSSREFS	`Family: A217831 (Euclid's triangle), A372728 (Kronecker's triangle), A372726 (Legendre's triangle), A373223 (Gauss' triangle).` `Cf. A055034 (row sums of abs values), A372876 (row sums).` `Cf. A055088.` `Sequence in context: A289011 A289071 A287931 * A289074 A289242 A346218` `Adjacent sequences: A372874 A372875 A372876 * A372878 A372879 A372880`
	KEYWORD	`sign,tabl`
	AUTHOR	`Peter Luschny, May 18 2024`
	STATUS	`approved`