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

%I A117908 #18 Sep 08 2022 08:45:24
%S A117908 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,
%T A117908 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,
%U A117908 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
%N A117908 Chequered (or checkered) triangle for odd prime p=3.
%C A117908 Row sums are A117909.
%C A117908 Diagonal sums are A117910.
%C A117908 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.
%H A117908 G. C. Greubel, <a href="/A117908/b117908.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A117908 G.f.: (1 +x*(1+y) +x^3*y)/((1-x^3)*(1-x^3*y^3)).
%F A117908 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.
%F A117908 T(n, k) = 1 if (n mod 3) < 2 and (k mod 3) < 2, otherwise 0. - _Kevin Ryde_, Oct 21 2021
%e A117908 Triangle begins
%e A117908   1;
%e A117908   1, 1;
%e A117908   0, 0, 0;
%e A117908   1, 1, 0, 1;
%e A117908   1, 1, 0, 1, 1;
%e A117908   0, 0, 0, 0, 0, 0;
%e A117908   1, 1, 0, 1, 1, 0, 1;
%e A117908   1, 1, 0, 1, 1, 0, 1, 1;
%e A117908   0, 0, 0, 0, 0, 0, 0, 0, 0;
%e A117908   1, 1, 0, 1, 1, 0, 1, 1, 0, 1;
%e A117908   1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1;
%e A117908   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
%e A117908   1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1;
%e A117908   1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1;
%e A117908   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
%t A117908 T[n_, k_]:= If[Abs[JacobiSymbol[Binomial[n, 2], 3] - 2*JacobiSymbol[Binomial[k, 2], 3]]==0, 1, 0];
%t A117908 Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 21 2021 *)
%o A117908 (Sage)
%o A117908 def A117908(n, k): return 1 if (n%3<2 and k%3<2) else 0
%o A117908 flatten([[A117908(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Oct 21 2021
%o A117908 (Magma)
%o A117908 A117908:= func< n,k | (n mod 3) lt 2 and (k mod 3) lt 2 select 1 else 0>;
%o A117908 [A117908(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Nov 18 2021
%Y A117908 Cf. A117904, A117909, A117910.
%K A117908 easy,nonn,tabl
%O A117908 0,1
%A A117908 _Paul Barry_, Apr 01 2006