A168030 Variant of pendular triangle A118340.
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, 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, 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
Offset: 0
Examples
Triangle begins as: 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, 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;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Magma
function t(n, k) // t = A118340 if k lt 0 or k gt n then return 0; elif k eq 0 then return 1; elif n gt 2*k then return t(n, n-k) + t(n-1, k); else return t(n, n-k-1) + t(n-1, k); end if; return t; end function; T:= func< n,k | t(n,k) mod 2 >; // A168030 [T(n,k): k in [0..n], n in [0..15]];
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Mathematica
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[n_, k_, p_]:= Mod[t[n,k,p], 2]; (* A168030 *) Table[T[n,k,1], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 12 2023 *)
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SageMath
@CachedFunction def t(n, k): # t = A118340 if (k<0 or k>n): return 0 elif (k==0): return 1 elif (n>2*k): return t(n, n-k) + t(n-1, k) else: return t(n, n-k-1) + t(n-1, k) def A168030(n,k): return t(n,k)%2 flatten([[A168030(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jan 12 2023
Formula
From G. C. Greubel, Jan 12 2023: (Start)
T(n, k) = A118340(n, k) mod 2.
Sum_{k=0..n} T(n, k) = A168148(n). (End)
Comments