cp's OEIS Frontend

A024693 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.

%I A024693 #7 Jul 16 2022 03:53:39
%S A024693 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 A024693 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 A024693 4,5,5,5,5,5,4,5,5,5,5,5,5,5,5,4,5,5,4,5,5,4
%N A024693 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 A024693 G. C. Greubel, <a href="/A024693/b024693.txt">Table of n, a(n) for n = 1..5000</a>
%F A024693 a(n) = Sum_{k=1..floor((n+1)/2)} A023533(k)*A014306(n+1-k).
%F A024693 a(n) = Sum_{k=1..floor((n+1)/2)} A023533(k)*(1 - A023533(n-k+1)). - _G. C. Greubel_, Jul 15 2022
%t A024693 A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
%t A024693 A024693[n_]:= A024693[n]= Sum[(1-A023533[n-k+2])*A023533[k], {k,Floor[(n+1)/2]}];
%t A024693 Table[A024693[n], {n,0,100}] (* _G. C. Greubel_, Jul 15 2022 *)
%o A024693 (Magma)
%o A024693 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A024693 [(&+[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 A024693 (SageMath)
%o A024693 def A023533(n):
%o A024693     if binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n: return 0
%o A024693     else: return 1
%o A024693 [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 A024693 Cf. A014306, A023533.
%K A024693 nonn
%O A024693 1,8
%A A024693 _Clark Kimberling_