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A168030 Variant of pendular triangle A118340.

%I A168030 #8 Jan 12 2023 01:38:29
%S A168030 1,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0,1,
%T A168030 1,0,1,1,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,
%U A168030 0,0,1,0,0,0,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,1,1,1,0
%N A168030 Variant of pendular triangle A118340.
%C A168030 Replaced the sums (f(a,b) = a + b) by the operators f(a,b) = a^2 -a*b + b^2 in the construction of triangle in A118340.
%H A168030 G. C. Greubel, <a href="/A168030/b168030.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A168030 From _G. C. Greubel_, Jan 12 2023: (Start)
%F A168030 T(n, k) = A118340(n, k) mod 2.
%F A168030 Sum_{k=0..n} T(n, k) = A168148(n). (End)
%e A168030 Triangle begins as:
%e A168030   1;
%e A168030   1, 0;
%e A168030   1, 1, 0;
%e A168030   1, 0, 1, 0;
%e A168030   1, 1, 0, 1, 0;
%e A168030   1, 0, 1, 1, 1, 0;
%e A168030   1, 1, 1, 0, 0, 1, 0;
%e A168030   1, 0, 0, 0, 0, 1, 1, 0;
%e A168030   1, 1, 0, 1, 1, 1, 0, 1, 0;
%e A168030   1, 0, 1, 0, 0, 1, 1, 1, 1, 0;
%e A168030   1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0;
%t A168030 t[n_, k_, p_]:= t[n, k, p]= If[k<0 || k>n, 0, If[k==0, 1, If[n<=2*k, t[n,n-k-1,p] +p*t[n-1,k,p], t[n,n-k,p] +t[n-1,k, p]]]]; (* A118340 *)
%t A168030 T[n_, k_, p_]:= Mod[t[n,k,p], 2]; (* A168030 *)
%t A168030 Table[T[n,k,1], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 12 2023 *)
%o A168030 (Magma)
%o A168030 function t(n, k) // t = A118340
%o A168030   if k lt 0 or k gt n then return 0;
%o A168030   elif k eq 0 then return 1;
%o A168030   elif n gt 2*k then return t(n, n-k) + t(n-1, k);
%o A168030   else return t(n, n-k-1) + t(n-1, k);
%o A168030   end if; return t;
%o A168030 end function;
%o A168030 T:= func< n,k | t(n,k) mod 2 >; // A168030
%o A168030 [T(n,k): k in [0..n], n in [0..15]];
%o A168030 (SageMath)
%o A168030 @CachedFunction
%o A168030 def t(n, k): # t = A118340
%o A168030     if (k<0 or k>n): return 0
%o A168030     elif (k==0): return 1
%o A168030     elif (n>2*k): return t(n, n-k) + t(n-1, k)
%o A168030     else: return t(n, n-k-1) + t(n-1, k)
%o A168030 def A168030(n,k): return t(n,k)%2
%o A168030 flatten([[A168030(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Jan 12 2023
%Y A168030 Cf. A118340, A168148 (row sums).
%K A168030 nonn,tabl
%O A168030 0,1
%A A168030 _Philippe Deléham_, Nov 17 2009