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

József Beck, 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

G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened

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)];

Mathematica

Table[P = Prime[n]; Table[JacobiSymbol[k, P], {k, P-1}]//Accumulate, {n, 15}]// Flatten (* G. C. Greubel, Oct 05 2024 *)

Prog

(Magma)
A226519:= func< n, k | n eq 1 select k else  (&+[JacobiSymbol(j, NthPrime(n)): j in [0..k]]) >;
[A226519(n, k) : k in [1..NthPrime(n)-1], n in [1..15]]; // G. C. Greubel, Oct 05 2024
(SageMath)
def A226519(n, k): return k if n==1 else sum(jacobi_symbol(j, nth_prime(n)) for j in range(k+1))
flatten([[A226519(n, k) for k in range(1, nth_prime(n))] for n in range(1, 16)]) # G. C. Greubel, Oct 05 2024

Crossrefs

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

Cf. A165582, A226518.

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