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a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)s(n+1-k), where k = floor((n+1)/2), s = A023533.
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%I #11 Jan 08 2023 17:36:06

%S 1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,

%T 1,0,0,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,

%U 1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1

%N a(n) = s(1)s(n) + s(2)s(n-1) + ... + s(k)s(n+1-k), where k = floor((n+1)/2), s = A023533.

%H G. C. Greubel, <a href="/A024692/b024692.txt">Table of n, a(n) for n = 1..5000</a>

%F a(n) = Sum_{k=1..floor((n+1)/2)} A023533(k)*A023533(n-k+1).

%t A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];

%t A024692[n_]:= A024692[n]= Sum[A023533[k]*A023533[n+1-k], {k, Floor[(n+1)/2]}];

%t Table[A024692[n], {n,100}] (* _G. C. Greubel_, Jul 14 2022 *)

%o (Magma)

%o A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;

%o [(&+[A023533(k)*A023533(n-k+1): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // _G. C. Greubel_, Jul 14 2022

%o (SageMath)

%o def A023533(n):

%o if binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n: return 0

%o else: return 1

%o [sum(A023533(k)*A023533(n-k+1) for k in (1..((n+1)//2))) for n in (1..100)] # _G. C. Greubel_, Jul 14 2022

%Y Cf. A023533.

%K nonn

%O 1,139

%A _Clark Kimberling_