A372728 Kronecker's triangle read by rows. T(n, k) = K(n, k) where K(n, k) is the Kronecker symbol (n / k).

0, 1, 1, 0, 1, 0, 0, 1, -1, 0, 0, 1, 0, 1, 0, 0, 1, -1, -1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, -1, 1, 0, 0, 1, 0, -1, 0, -1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 1, -1, -1, 1, 1, 1, 1, -1, 1, -1, 0, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0

Offset

0

Comments

Kronecker's triangle is a signed version of Euclid's triangle A217831 and generalizes and incorporates the Legendre and Jacobi symbols. However, the definition domain of the latter two symbols is inconsistently defined, so the Legendre symbol is typically only defined if the second argument is an odd prime or at least a positive odd integer. All such restrictions do not apply here (apart from limiting the range to the triangular domain 0 <= k <= n).

A096398 lists the indices of the rows that are identical in both triangles (i.e., Kronecker's and Euclid's). These are exactly the rows of T without negative terms.

References

Henri Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, 1993, p. 28.

Links

Table of n, a(n) for n=0..90.

Jean-Paul Allouche, Leo Goldmakher, Mock characters and the Kronecker symbol, arXiv:1608.03957 [math.NT], 2016.

Diego F. Aranha, B. S. Hvass, Bas Spitters and Mehdi Tibouchi, Faster constant-time evaluation of the Kronecker symbol with application to elliptic curve hashing, ACM SIGSAC, Nov. 2023, p. 3228-3238.

Pierre Cartier, Sur une généralisation des symboles de Legendre-Jacobi, Enseign. Math. (2), 16 (1970), 31-48.

Leopold Kronecker, Zur Theorie der elliptischen Functionen, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, Jahrgang 1885, S. 770

Example

Triangle K(n, k) starts:
  [0] 0;
  [1] 1, 1;
  [2] 0, 1,  0;
  [3] 0, 1, -1,  0;
  [4] 0, 1,  0,  1, 0;
  [5] 0, 1, -1, -1, 1,  0;
  [6] 0, 1,  0,  0, 0,  1, 0;
  [7] 0, 1,  1,  1, 1, -1, 1, 0;
  [8] 0, 1,  0, -1, 0, -1, 0, 1, 0;
  [9] 0, 1,  1,  0, 1,  1, 0, 1, 1, 0;
.
Not limiting the range of k leads to the square array:
  [0] 0,  1,  0,  0,  0,  0,  0,  0,  0,  0, ...  A063524
  [1] 1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...  A000012
  [2] 0,  1,  0, -1,  0, -1,  0,  1,  0,  1, ...  A091337
  [3] 0,  1, -1,  0,  1, -1,  0, -1, -1,  0, ...  A091338
  [4] 0,  1,  0,  1,  0,  1,  0,  1,  0,  1, ...  A000035
  [5] 0,  1, -1, -1,  1,  0,  1, -1, -1,  1, ...  A080891
  [6] 0,  1,  0,  0,  0,  1,  0, -1,  0,  0, ...  A322796
  [7] 0,  1,  1,  1,  1, -1,  1,  0,  1,  1, ...
  [8] 0,  1,  0, -1,  0, -1,  0,  1,  0,  1, ...
  [9] 0,  1,  1,  0,  1,  1,  0,  1,  1,  0, ...
  ...

Maple

K := (n, k) -> NumberTheory:-KroneckerSymbol(n, k):
seq(seq(K(n, k), k = 0..n), n = 0..12);

Mathematica

Table[KroneckerSymbol[n, k], {n, 0, 12}, {k, 0, n}] // Flatten

Prog

(Python) # Demonstrates how the Kronecker symbol
# can be layered on the Jacobi symbol.
from sympy import igcd
from sympy.ntheory import jacobi_symbol
# Alternatively, you can use the function JacobiSymbol from A372877.
def is_even(n): return n % 2 == 0
def kronecker_symbol(n, k):
    if not (igcd(n, k) == 1): return 0
    if n == 1 or k == 1: return 1
    if is_even(k):
        if is_even(n): return 0
        s = 1 if is_even((n + 1) // 4) else -1
        if k == 2: return s
        return s * kronecker_symbol(n, k // 2)
    return jacobi_symbol(n, k)
for n in range(20):
    print([kronecker_symbol(n, k) for k in range(n + 1)])

Crossrefs

Family: A217831 (Euclid's triangle), A372877 (Jacobi's triangle), A372726 (Legendre's triangle), A373223 (Gauss' triangle).

Cf. A071961 (row sums), A000010 (row sum of absolute terms, for n >= 2), A063524 (column 0 and main diagonal).

Cf. A020639 (The index of the first appearance of 0 in row n of the array.)

Cf. A373180 (The index of the first appearance of 1 in row n of the array.)

Cf. A373088 (The index of the first appearance of -1 in row n of the array.)

Cf. A096396 (number of 1s in the triangle row n), A096397 (number of -1s in the triangle row n), A062830 (number of 0s in the triangle row n).

Cf. A096398, A372519 (indices of the rows whose sum vanishes).

Cf. A063524, A000012, A091337, A091338, A000035, A080891, A322796.

Sequence in context: A189084 A143222 A286490 * A217831 A010060 A316569

Adjacent sequences: A372725 A372726 A372727 * A372729 A372730 A372731

Keyword

sign,tabl

Author

Peter Luschny, May 16 2024

Status

approved