A023533 a(n) = 1 if n is of the form m(m+1)(m+2)/6, and 0 otherwise.

1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Clark Kimberling, Jun 14 1998

Keywords

• nonn
• easy

Comments

a(n) is the characteristic function of tetrahedral numbers (A000292). - Mikael Aaltonen, Mar 28 2015

Links

• Danny Rorabaugh, Table of n, a(n) for n = 0..10000
• Index entries for characteristic functions.

Crossrefs

Cf. A000292, A014306, A145397.

Programs

Mathematica

With[{ms=Table[m(m+1)(m+2)/6,{m,0,20}]},Table[If[MemberQ[ms,n],1,0], {n,0,100}]] (* Harvey P. Dale, Jul 25 2011 *)
a[n_]:=Boole[Binomial[Floor[(6n-1)^(1/3)]+2, 3] == n]; Array[a,99,0] (* Stefano Spezia, Sep 15 2024 after Michel Marcus, Jul 19 2022 *)

PARI

lista(nn) = {v = vector(nn); for (n=0, nn, i = 1+n*(n+1)*(n+2)/6; if (i > nn, break); v[i] = 1;); v;} \\ Michel Marcus, Mar 16 2015

PARI

a(n) = if(n==0, return(1)); my(t = sqrtnint(6*n-1, 3)); binomial(t+2, 3) == n; \\ Michel Marcus, Jul 19 2022; after A014306

Sage

# Generates an array with at least N terms.
def A023533_list(N):
    A = []
    for m in range(ceil((6*N)^(1/3))):
        A.extend([0]*(binomial(m+2, 3) - len(A)) + [1])
    return A
print(A023533_list(40))
# Danny Rorabaugh, Mar 16 2015

Formula

a(A000292(n))=1; a(A145397(n))=0; a(n)=1-A014306(n). - Reinhard Zumkeller, Oct 14 2008

For n > 0, a(n) = floor(t(n+1) + 1/(3 * t(n+1)) - 1) - floor(t(n) + 1/(3 * t(n)) - 1), where t(n) = ( sqrt(243*n^2-1)/3^(3/2) + 3*n )^(1/3). - Mikael Aaltonen, Mar 28 2015; corrected by Michel Marcus, Jul 17 2022