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A023604 Convolution of A023532 and A023533.

%I A023604 #7 Jul 16 2022 03:46:50
%S A023604 1,0,1,2,0,2,2,1,1,3,2,2,3,1,3,3,2,2,3,3,3,4,2,3,4,4,3,3,3,3,4,4,3,4,
%T A023604 4,3,5,4,3,5,5,5,4,3,5,4,4,4,5,5,5,5,4,2,5,6,4,6,6,5,5,6,4,5,5,6,6,5,
%U A023604 4,6,6,6,5,5,5,6,5,5,6,5,6,5,6,6,6,6,7,5,7,5
%N A023604 Convolution of A023532 and A023533.
%H A023604 G. C. Greubel, <a href="/A023604/b023604.txt">Table of n, a(n) for n = 1..5000</a>
%F A023604 From _G. C. Greubel_, Jul 16 2022: (Start)
%F A023604 a(n) = Sum_{j=1..n} A023532(n-j+1) * A023533(j).
%F A023604 a(n) = Sum_{j=1..n} (1 - A023531(n-j+1)) * A023533(j). (End)
%t A023604 A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
%t A023604 A023531[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2], 1, 0];
%t A023604 A023604[n_]:= A023604[n]= Sum[A023533[k]*(1-A023531[n-k+1]), {k,n}];
%t A023604 Table[A023604[n], {n,100}] (* _G. C. Greubel_, Jul 16 2022 *)
%o A023604 (Magma)
%o A023604 A023532:= func< n | IsIntegral((Sqrt(8*n+9) - 3)/2) select 0 else 1 >;
%o A023604 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A023604 [(&+[A023533(k)*A023532(n+1-k): k in [1..n]]): n in [1..100]]; // _G. C. Greubel_, Jul 16 2022
%o A023604 (SageMath)
%o A023604 def A023532(n): return 0 if ((sqrt(8*n+9) -3)/2).is_integer() else 1
%o A023604 @CachedFunction
%o A023604 def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1
%o A023604 [sum(A023533(k)*A023532(n-k+1) for k in (1..n)) for n in (1..100)] # _G. C. Greubel_, Jul 16 2022
%Y A023604 Cf. A023531, A023532, A023533.
%K A023604 nonn
%O A023604 1,4
%A A023604 _Clark Kimberling_