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A024690 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.

%I A024690 #6 Sep 07 2022 18:12:47
%S A024690 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 A024690 23,26,28,31,34,36,39,43,49,54,59,65,15,18,20,23,26,28,31,34,36,39,41,
%U A024690 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 A024690 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 A024690 G. C. Greubel, <a href="/A024690/b024690.txt">Table of n, a(n) for n = 1..5000</a>
%t A024690 A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
%t A024690 A001950[n_]:= Floor[n*GoldenRatio^2];
%t A024690 A024690[n_]:= A024690[n]= Sum[A001950[j]*A023533[n-j+1], {j, Floor[(n+1)/2]}];
%t A024690 Table[A024690[n], {n, 130}] (* _G. C. Greubel_, Sep 07 2022 *)
%o A024690 (Magma)
%o A024690 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A024690 A001950:= func< n |  Floor(n*(3+Sqrt(5))/2) >;
%o A024690 A024690:= func< n | (&+[A001950(k)*A023533(n+1-k): k in [1..Floor((n+1)/2)]]) >;
%o A024690 [A024690(n): n in [1..130]]; // _G. C. Greubel_, Sep 07 2022
%o A024690 (SageMath)
%o A024690 @CachedFunction
%o A024690 def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1
%o A024690 def A001950(n): return floor(n*golden_ratio^2)
%o A024690 def A024690(n): return sum(A001950(k)*A023533(n-k+1) for k in (1..((n+1)//2)))
%o A024690 [A024690(n) for n in (1..130)] # _G. C. Greubel_, Sep 07 2022
%Y A024690 Cf. A001950, A023533.
%K A024690 nonn
%O A024690 1,1
%A A024690 _Clark Kimberling_