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
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
Programs
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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
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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
Comments