cp's OEIS Frontend

A024687 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A000201 (lower Wythoff sequence), t = A023533.

%I A024687 #7 Aug 02 2022 09:13:12
%S A024687 1,0,0,1,3,4,6,0,0,1,3,4,6,8,9,11,12,14,16,1,3,4,6,8,9,11,12,14,16,17,
%T A024687 19,21,22,24,26,30,33,36,40,9,11,12,14,16,17,19,21,22,24,25,27,29,30,
%U A024687 32,33,36,40,42,46,50,52,56,58,62,66,68,72,76,78,24,25,27,29,30,32,33,35,37,38,40
%N A024687 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A000201 (lower Wythoff sequence), t = A023533.
%H A024687 G. C. Greubel, <a href="/A024687/b024687.txt">Table of n, a(n) for n = 1..5000</a>
%F A024687 a(n) = Sum_{k=1..floor((n+1)/2)} A000201(k) * A023533(n-k+1).
%t A024687 A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1,3]] +2, 3]!= n, 0,1];
%t A024687 A024687[n_]:= A024687[n]= Sum[Floor[j*GoldenRatio]*A023533[n-j+1], {j, Floor[(n+ 1)/2]}];
%t A024687 Table[A024687[n], {n, 100}] (* _G. C. Greubel_, Aug 01 2022 *)
%o A024687 (Magma)
%o A024687 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A024687 [(&+[Floor(k*(1+Sqrt(5))/2)*A023533(n+1-k): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // _G. C. Greubel_, Aug 01 2022
%o A024687 (SageMath)
%o A024687 @CachedFunction
%o A024687 def A023533(n): return 0 if (binomial(floor((6*n-1)^(1/3)) +2, 3)!= n) else 1
%o A024687 def A024687(n): return sum(floor(j*golden_ratio)*A023533(n-j+1) for j in (1..((n+1)//2)))
%o A024687 [A024687(n) for n in (1..100)] # _G. C. Greubel_, Aug 01 2022
%Y A024687 Cf. A000201, A023533.
%K A024687 nonn
%O A024687 1,5
%A A024687 _Clark Kimberling_