A056556 -id:A056556 - cp's OEIS frontend

A023671 Convolution of A023533 and A014306.

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, 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, 5, 6, 6, 6, 6, 6, 4, 6, 6, 6, 6, 6, 6, 6, 6, 5, 7, 7, 5, 7, 7, 5

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A014306, A023533, A023670, A056556,

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[(1-A023533(k))*A023533(n+1-k): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jul 18 2022

A023533[n_]:= If[Binomial[Floor[Surd[6*n-1, 3]] + 2, 3] != n, 0, 1];
A023671[n_]:= A023613[n]= Sum[(1-A023533[k])*A023533[n-k+1], {k,n}];
Table[A023671[n], {n, 100}] (* G. C. Greubel, Jul 18 2022 *)

@CachedFunction
def A023533(n): return 1 if binomial(floor((6*n-1)^(1/3)) +2, 3)!=n else 0
def A023671(n): return sum((1-A023533(k))*A023533(n-k+1) for k in (1..n))
[A023671(n) for n in (1..100)] # G. C. Greubel, Jul 18 2022

a(n) = Sum_{j=1..n} A023533(n-j+1)*A014306(j).
From G. C. Greubel, Jul 18 2022: (Start)
a(n) = Sum_{j=1..n} A023533(n-j+1)*(1 - A023533(j)).
a(n) = A056556(n) - A023670(n). (End)

A332616 a(n) = value of the cubic form A^3 + B^3 + C^3 - 3ABC evaluated at row n of the table in A331195.

Original entry on oeis.org

0, 1, 2, 0, 8, 9, 4, 16, 5, 0, 27, 28, 20, 35, 18, 7, 54, 28, 8, 0, 64, 65, 54, 72, 49, 32, 91, 56, 27, 10, 128, 81, 40, 11, 0, 125, 126, 112, 133, 104, 81, 152, 108, 70, 44, 189, 130, 77, 36, 13, 250, 176, 108, 52, 14, 0, 216, 217, 200, 224, 189, 160, 243

Offset: 0

Mehmet A. Ates, Jun 08 2020

No term in the sequence is congruent to 3 or 6 (mod 9).

			For n=3, a(n) = f[1,1,0] = 1^3 + 1^3 + 0^3 - 3*1*1*0 = 2.

Mehmet A. Ates, Table of n, a(n) for n = 0..19599
Mathematical Association of America, 2019 William Lowell Putnam Mathematical Competition Problems

Cf. A056556, A056557, A056558.
Cf. A330013, A331195.
Cf. A074232 (in ascending order, strictly positive & without duplicates).

SeqSize = 30;
ListSize = 120;
F3List = List[];
f3[a_, b_, c_] := a^3 + b^3 + c^3 - 3*a*b*c
For[i = 0, i <= SeqSize, i++, For[j = 0, j <= i, j++, For[k = 0, k <= j, k++, AppendTo[F3List, f3[i, j, k]]]]]
ListPlot[F3List, PlotLabel -> "a(n)"]
Print["First ", ListSize, " elements of a(n): ", Take[F3List, ListSize]]

a(n) = A056556(n)^3 + A056557(n)^3 + A056558(n)^3 - 3*A056556(n)*A056557(n)*A056558(n).

Edited by N. J. A. Sloane, Aug 06 2020

cp's OEIS Frontend

A023671 Convolution of A023533 and A014306.

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