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%I #7 Jul 16 2022 03:45:01
%S 0,1,0,0,2,0,0,1,1,0,1,1,0,2,0,0,1,1,0,1,1,0,2,1,0,0,1,1,1,1,0,0,1,0,
%T 1,2,0,1,2,0,0,0,1,2,0,1,1,1,0,0,0,0,1,3,0,0,2,0,0,1,1,0,2,1,1,0,0,1,
%U 2,0,0,0,1,1,1,0,1,1,0,1,0,1,0,1,1,1,0,2,0,2
%N Convolution of A023531 and A023533.
%H G. C. Greubel, <a href="/A023565/b023565.txt">Table of n, a(n) for n = 1..5000</a>
%F a(n) = Sum_{j=1..n} A023533(j) * A023531(n-j+1). - _G. C. Greubel_, Jul 16 2022
%t A023531[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2], 1, 0];
%t A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
%t A023565[n_]:= A023565[n]= Sum[A023533[k]*A023531[n-k+1], {k,n}];
%t Table[A023565[n], {n,100}] (* _G. C. Greubel_, Jul 16 2022 *)
%o (Magma)
%o A023531:= func< n | IsIntegral((Sqrt(8*n+9) - 3)/2) select 1 else 0 >;
%o A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o [(&+[A023533(k)*A023531(n+1-k): k in [1..n]]): n in [1..100]]; // _G. C. Greubel_, Jul 16 2022
%o (SageMath)
%o @CachedFunction
%o def A023531(n): return 1 if ((sqrt(8*n+9) -3)/2).is_integer() else 0
%o @CachedFunction
%o def A023533(n): return 0 if binomial( floor((6*n-1)^(1/3)) +2, 3)!=n else 1
%o [sum(A023533(k)*A023531(n-k+1) for k in (1..n)) for n in (1..100)] # _G. C. Greubel_, Jul 16 2022
%Y Cf. A023531, A023533.
%K nonn
%O 1,5
%A _Clark Kimberling_
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