A226519 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} Legendre(i,prime(n)).

1, 1, 0, 1, 0, -1, 0, 1, 2, 1, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, -1, -2, -1, 0, -1, 0, 1, 2, 1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, -2, -1, 0, 1, 0, -1, 0, 1, 2, 3, 2, 3, 2, 3, 2, 1, 0, -1, 0, 1, 0, 1, 2, 3, 4, 3, 4, 3, 4, 5, 4, 3, 4, 5, 4, 3, 4, 3, 4, 3, 2, 1, 0

Offset

1,9

Comments

Strictly speaking, the symbol in the definition is the Legendre-Jacobi-Kronecker symbol, since the Legendre symbol is defined only for odd primes.

References

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.

Links

Table of n, a(n) for n=1..91.

Example

Triangle begins:
[1],
[1, 0],
[1, 0, -1, 0],
[1, 2, 1, 2, 1, 0],
[1, 0, 1, 2, 3, 2, 1, 0, 1, 0],
[1, 0, 1, 2, 1, 0, -1, -2, -1, 0, -1, 0],
[1, 2, 1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, -2, -1, 0],
...

Maple

with(numtheory);
T:=(n, k)->add(legendre(i, ithprime(n)), i=1..k);
f:=n->[seq(T(n, k), k=1..ithprime(n)-1)];
[seq(f(n), n=1..15)];

Crossrefs

Cf. A165582. A variant of A226518, which is the main entry for this triangle.

Sequence in context: A107016 A318702 A360536 * A066057 A060588 A221167

Adjacent sequences: A226516 A226517 A226518 * A226520 A226521 A226522

Keyword

sign,tabf

Author

N. J. A. Sloane, Jun 19 2013

Status

approved