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a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023533, t = A014306.
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%I #7 Jul 16 2022 03:53:39

%S 0,1,1,0,1,1,1,2,2,1,2,2,1,2,2,2,2,2,2,2,3,3,2,3,3,3,3,3,2,3,3,3,3,3,

%T 2,3,3,2,3,4,4,4,4,3,4,4,4,4,4,4,4,4,4,3,4,3,4,4,3,4,4,4,4,4,3,4,4,4,

%U 4,5,5,5,5,5,4,5,5,5,5,5,5,5,5,4,5,5,4,5,5,4

%N a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023533, t = A014306.

%H G. C. Greubel, <a href="/A024693/b024693.txt">Table of n, a(n) for n = 1..5000</a>

%F a(n) = Sum_{k=1..floor((n+1)/2)} A023533(k)*A014306(n+1-k).

%F a(n) = Sum_{k=1..floor((n+1)/2)} A023533(k)*(1 - A023533(n-k+1)). - _G. C. Greubel_, Jul 15 2022

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

%t A024693[n_]:= A024693[n]= Sum[(1-A023533[n-k+2])*A023533[k], {k,Floor[(n+1)/2]}];

%t Table[A024693[n], {n,0,100}] (* _G. C. Greubel_, Jul 15 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)*(1-A023533(n+1-k)): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // _G. C. Greubel_, Jul 15 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)*(1-A023533(n-k+1)) for k in (1..((n+1)//2))) for n in (1..100)] # _G. C. Greubel_, Jul 15 2022

%Y Cf. A014306, A023533.

%K nonn

%O 1,8

%A _Clark Kimberling_