A025075 - OEIS

%I #7 Aug 03 2022 02:38:49

%S 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 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 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 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 G. C. Greubel, <a href="/A025075/b025075.txt">Table of n, a(n) for n = 1..5000</a>

%F a(n)_= Sum_{k=1..floor((n+1)/2)} A023532(k) * A023533(n-k+2).

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

%t A023532[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2],0,1];

%t A025075[n_]:= A025075[n]= Sum[A023532[j]*A023533[n-j+2], {j, Floor[(n+1)/2]}];

%t Table[A025075[n], {n,130}] (* _G. C. Greubel_, Aug 02 2022 *)

%o (Magma)

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

%o A023532:= func< n |  IsSquare(8*n+9) select 0 else 1 >;

%o A025075:= func< n | (&+[A023532(k)*A023533(n+2-k): k in [1..Floor((n+1)/2)]]) >;

%o [A025075(n): n in [1..130]]; // _G. C. Greubel_, Aug 02 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 A023532(n): return 0 if is_square(8*n+9) else 1

%o def A025075(n): return sum(A023532(k)*A023533(n-k+2) for k in (1..((n+1)//2)))

%o [A025075(n) for n in (1..130)] # _G. C. Greubel_, Aug 02 2022

%Y Cf. A023532, A023533.

%K nonn

%O 1,34

%A _Clark Kimberling_