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A023671 Convolution of A023533 and A014306.

%I A023671 #11 Aug 05 2024 06:27:03
%S A023671 0,1,1,0,2,2,1,2,2,1,3,3,1,3,3,3,3,3,2,2,4,4,2,4,4,4,4,4,2,4,4,4,4,4,
%T A023671 3,5,5,3,4,5,5,5,5,3,5,5,5,5,5,5,5,5,5,3,5,4,6,6,4,6,6,6,6,6,4,6,6,6,
%U A023671 5,6,6,6,6,6,4,6,6,6,6,6,6,6,6,5,7,7,5,7,7,5
%N A023671 Convolution of A023533 and A014306.
%H A023671 G. C. Greubel, <a href="/A023671/b023671.txt">Table of n, a(n) for n = 1..5000</a>
%F A023671 a(n) = Sum_{j=1..n} A023533(n-j+1)*A014306(j).
%F A023671 From _G. C. Greubel_, Jul 18 2022: (Start)
%F A023671 a(n) = Sum_{j=1..n} A023533(n-j+1)*(1 - A023533(j)).
%F A023671 a(n) = A056556(n) - A023670(n). (End)
%t A023671 A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] + 2, 3] != n, 0, 1];
%t A023671 A023671[n_]:= A023613[n]= Sum[(1-A023533[k])*A023533[n-k+1], {k,n}];
%t A023671 Table[A023671[n], {n, 100}] (* _G. C. Greubel_, Jul 18 2022 *)
%o A023671 (Magma)
%o A023671 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A023671 [(&+[(1-A023533(k))*A023533(n+1-k): k in [1..n]]): n in [1..100]]; // _G. C. Greubel_, Jul 18 2022
%o A023671 (Sage)
%o A023671 @CachedFunction
%o A023671 def A023533(n): return 1 if binomial(floor((6*n-1)^(1/3)) +2, 3)!=n else 0
%o A023671 def A023671(n): return sum((1-A023533(k))*A023533(n-k+1) for k in (1..n))
%o A023671 [A023671(n) for n in (1..100)] # _G. C. Greubel_, Jul 18 2022
%Y A023671 Cf. A014306, A023533, A023670, A056556,
%K A023671 nonn
%O A023671 1,5
%A A023671 _Clark Kimberling_