A117898 Number triangle 2^abs(L(C(n,2)/3) - L(C(k,2)/3))*[k<=n] where L(j/p) is the Legendre symbol of j and p.

1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1

Offset

0,4

Comments

Row sums are A117899. Diagonal sums are A117900. Inverse is A117901. A117898 mod 2 is A117904.

Links

G. C. Greubel, Table of n, a(n) for the first 101 rows, flattened

Formula

G.f.: (1 +x*(1+y) +x^2*(2+2*y+y^2) +x^3*y(1+2*y) +2*x^4*y^2)/((1-x^3)*(1-x^3*y^3)).

T(n, k) = [k<=n]*2^abs(L(C(n,2)/3) - L(C(k,2)/3)).

Example

Triangle begins
  1;
  1, 1;
  2, 2, 1;
  1, 1, 2, 1;
  1, 1, 2, 1, 1;
  2, 2, 1, 2, 2, 1;
  1, 1, 2, 1, 1, 2, 1;
  1, 1, 2, 1, 1, 2, 1, 1;
  2, 2, 1, 2, 2, 1, 2, 2, 1;
  1, 1, 2, 1, 1, 2, 1, 1, 2, 1;
  1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1;
  2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1;
  1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1;

Mathematica

Flatten[CoefficientList[CoefficientList[Series[(1 +x(1+y) +x^2(2+2y+y^2) +x^3*y(1 +2y) +2x^4*y^2)/((1-x^3)(1-x^3*y^3)), {x, 0, 15}, {y, 0, 15}], x], y]] (* G. C. Greubel, May 03 2017 *)
T[n_, k_]:= 2^Abs[JacobiSymbol[Binomial[n, 2], 3] - JacobiSymbol[Binomial[k, 2], 3]]; Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 27 2021 *)

Prog

(Sage)
def T(n, k): return 2^abs(kronecker(binomial(n, 2), 3) - kronecker(binomial(k, 2), 3))
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 27 2021

Crossrefs

Cf. A117898, A117899, A117900, A117901, A117904.

Sequence in context: A239110 A278514 A243840 * A212810 A072344 A140500

Adjacent sequences: A117895 A117896 A117897 * A117899 A117900 A117901

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 01 2006

Status

approved