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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.

%I A117898 #9 Oct 01 2021 13:10:42
%S A117898 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,
%T A117898 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,
%U A117898 1,2,2,1,2,2,1,2,2,1,1,1,2,1,1,2,1,1,2,1,1,2,1
%N 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.
%C A117898 Row sums are A117899. Diagonal sums are A117900. Inverse is A117901. A117898 mod 2 is A117904.
%H A117898 G. C. Greubel, <a href="/A117898/b117898.txt">Table of n, a(n) for the first 101 rows, flattened</a>
%F A117898 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)).
%F A117898 T(n, k) = [k<=n]*2^abs(L(C(n,2)/3) - L(C(k,2)/3)).
%e A117898 Triangle begins
%e A117898   1;
%e A117898   1, 1;
%e A117898   2, 2, 1;
%e A117898   1, 1, 2, 1;
%e A117898   1, 1, 2, 1, 1;
%e A117898   2, 2, 1, 2, 2, 1;
%e A117898   1, 1, 2, 1, 1, 2, 1;
%e A117898   1, 1, 2, 1, 1, 2, 1, 1;
%e A117898   2, 2, 1, 2, 2, 1, 2, 2, 1;
%e A117898   1, 1, 2, 1, 1, 2, 1, 1, 2, 1;
%e A117898   1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1;
%e A117898   2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1;
%e A117898   1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1;
%t A117898 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 A117898 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 *)
%o A117898 (Sage)
%o A117898 def T(n, k): return 2^abs(kronecker(binomial(n,2), 3) - kronecker(binomial(k,2), 3))
%o A117898 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Sep 27 2021
%Y A117898 Cf. A117898, A117899, A117900, A117901, A117904.
%K A117898 easy,nonn,tabl
%O A117898 0,4
%A A117898 _Paul Barry_, Apr 01 2006