cp's OEIS Frontend

A025075 a(n) = s(1)*t(n+1) + s(2)*t(n) + ... + s(k)*t(n-k+2), where k = floor((n+1)/2), s = A023532, t = A023533.

%I A025075 #7 Aug 03 2022 02:38:49
%S A025075 0,0,1,0,1,0,0,0,1,0,1,1,0,1,1,1,0,0,1,0,1,1,0,1,1,1,0,1,1,1,1,0,1,2,
%T A025075 1,2,2,0,1,1,1,0,1,1,1,1,0,1,1,1,1,1,0,1,2,1,2,2,1,1,2,2,1,2,2,2,2,0,
%U A025075 1,1,1,1,1,0,1,1,1,1,1,1,0,1,2,1,2,2,1,2,1,2,1,2,2,2,2,1,2,1,2,2,2,1,2,2,2,2,2,1,1,1,1
%N A025075 a(n) = s(1)*t(n+1) + s(2)*t(n) + ... + s(k)*t(n-k+2), where k = floor((n+1)/2), s = A023532, t = A023533.
%H A025075 G. C. Greubel, <a href="/A025075/b025075.txt">Table of n, a(n) for n = 1..5000</a>
%F A025075 a(n)_= Sum_{k=1..floor((n+1)/2)} A023532(k) * A023533(n-k+2).
%t A025075 A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1,3]] +2,3]!= n,0,1];
%t A025075 A023532[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2],0,1];
%t A025075 A025075[n_]:= A025075[n]= Sum[A023532[j]*A023533[n-j+2], {j, Floor[(n+1)/2]}];
%t A025075 Table[A025075[n], {n,130}] (* _G. C. Greubel_, Aug 02 2022 *)
%o A025075 (Magma)
%o A025075 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A025075 A023532:= func< n |  IsSquare(8*n+9) select 0 else 1 >;
%o A025075 A025075:= func< n | (&+[A023532(k)*A023533(n+2-k): k in [1..Floor((n+1)/2)]]) >;
%o A025075 [A025075(n): n in [1..130]]; // _G. C. Greubel_, Aug 02 2022
%o A025075 (SageMath)
%o A025075 @CachedFunction
%o A025075 def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1
%o A025075 def A023532(n): return 0 if is_square(8*n+9) else 1
%o A025075 def A025075(n): return sum(A023532(k)*A023533(n-k+2) for k in (1..((n+1)//2)))
%o A025075 [A025075(n) for n in (1..130)] # _G. C. Greubel_, Aug 02 2022
%Y A025075 Cf. A023532, A023533.
%K A025075 nonn
%O A025075 1,34
%A A025075 _Clark Kimberling_