A117908 Chequered (or checkered) triangle for odd prime p=3.

1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1

Offset: 0

Author

Paul Barry, Apr 01 2006

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A117909.

Diagonal sums are A117910.

For odd prime p, T(n,k;p) = [k<=n]*0^abs(L(C(n,p-1)/p) - 2*L(C(k,p-1)/p)) defines a checkered triangle for p.

Examples

			Triangle begins
  1;
  1, 1;
  0, 0, 0;
  1, 1, 0, 1;
  1, 1, 0, 1, 1;
  0, 0, 0, 0, 0, 0;
  1, 1, 0, 1, 1, 0, 1;
  1, 1, 0, 1, 1, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0;
  1, 1, 0, 1, 1, 0, 1, 1, 0, 1;
  1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1;
  1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A117904, A117909, A117910.

Programs

Magma

A117908:= func< n,k | (n mod 3) lt 2 and (k mod 3) lt 2 select 1 else 0>;
[A117908(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Nov 18 2021

Mathematica

T[n_, k_]:= If[Abs[JacobiSymbol[Binomial[n, 2], 3] - 2*JacobiSymbol[Binomial[k, 2], 3]]==0, 1, 0];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 21 2021 *)

Sage

def A117908(n, k): return 1 if (n%3<2 and k%3<2) else 0
flatten([[A117908(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Oct 21 2021

Formula

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

T(n,k) = [k<=n] * 0^abs(L(C(n,2)/3) - 2*L(C(k,2)/3)) where L(j/p) is the Legendre symbol of j and p.

T(n, k) = 1 if (n mod 3) < 2 and (k mod 3) < 2, otherwise 0. - Kevin Ryde, Oct 21 2021