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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 = A001950 (upper Wythoff sequence), t = A023533.
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%I #6 Sep 07 2022 18:12:47

%S 2,0,0,2,5,7,10,0,0,2,5,7,10,13,15,18,20,23,26,2,5,7,10,13,15,18,20,

%T 23,26,28,31,34,36,39,43,49,54,59,65,15,18,20,23,26,28,31,34,36,39,41,

%U 44,47,49,52,54,59,65,69,75,81,85,91,95,101,107,111,117,123,127,39,41,44,47,49,52

%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 = A001950 (upper Wythoff sequence), t = A023533.

%H G. C. Greubel, <a href="/A024690/b024690.txt">Table of n, a(n) for n = 1..5000</a>

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

%t A001950[n_]:= Floor[n*GoldenRatio^2];

%t A024690[n_]:= A024690[n]= Sum[A001950[j]*A023533[n-j+1], {j, Floor[(n+1)/2]}];

%t Table[A024690[n], {n, 130}] (* _G. C. Greubel_, Sep 07 2022 *)

%o (Magma)

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

%o A001950:= func< n | Floor(n*(3+Sqrt(5))/2) >;

%o A024690:= func< n | (&+[A001950(k)*A023533(n+1-k): k in [1..Floor((n+1)/2)]]) >;

%o [A024690(n): n in [1..130]]; // _G. C. Greubel_, Sep 07 2022

%o (SageMath)

%o @CachedFunction

%o def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1

%o def A001950(n): return floor(n*golden_ratio^2)

%o def A024690(n): return sum(A001950(k)*A023533(n-k+1) for k in (1..((n+1)//2)))

%o [A024690(n) for n in (1..130)] # _G. C. Greubel_, Sep 07 2022

%Y Cf. A001950, A023533.

%K nonn

%O 1,1

%A _Clark Kimberling_