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A023604
Convolution of A023532 and A023533.
1
1, 0, 1, 2, 0, 2, 2, 1, 1, 3, 2, 2, 3, 1, 3, 3, 2, 2, 3, 3, 3, 4, 2, 3, 4, 4, 3, 3, 3, 3, 4, 4, 3, 4, 4, 3, 5, 4, 3, 5, 5, 5, 4, 3, 5, 4, 4, 4, 5, 5, 5, 5, 4, 2, 5, 6, 4, 6, 6, 5, 5, 6, 4, 5, 5, 6, 6, 5, 4, 6, 6, 6, 5, 5, 5, 6, 5, 5, 6, 5, 6, 5, 6, 6, 6, 6, 7, 5, 7, 5
OFFSET
1,4
LINKS
FORMULA
From G. C. Greubel, Jul 16 2022: (Start)
a(n) = Sum_{j=1..n} A023532(n-j+1) * A023533(j).
a(n) = Sum_{j=1..n} (1 - A023531(n-j+1)) * A023533(j). (End)
MATHEMATICA
A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
A023531[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2], 1, 0];
A023604[n_]:= A023604[n]= Sum[A023533[k]*(1-A023531[n-k+1]), {k, n}];
Table[A023604[n], {n, 100}] (* G. C. Greubel, Jul 16 2022 *)
PROG
(Magma)
A023532:= func< n | IsIntegral((Sqrt(8*n+9) - 3)/2) select 0 else 1 >;
A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[A023533(k)*A023532(n+1-k): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jul 16 2022
(SageMath)
def A023532(n): return 0 if ((sqrt(8*n+9) -3)/2).is_integer() else 1
@CachedFunction
def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1
[sum(A023533(k)*A023532(n-k+1) for k in (1..n)) for n in (1..100)] # G. C. Greubel, Jul 16 2022
CROSSREFS
KEYWORD
nonn
STATUS
approved