A189017 Zero-one sequence based on tetrahedral numbers: a(A000292(k))=a(k); a(A145397(k))=1-a(k); a(1)=0.

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

Offset

1

Links

Table of n, a(n) for n=1..135.

Example

Let u=A000292 and v=A145397, so that u(n)=n(n+1)(n+2)/6 and v=complement(u) for n>=1.  Then a is a self-generating zero-one sequence with initial value a(1)=0 and a(u(k))=a(k); a(v(k))=1-a(k).

Mathematica

u[n_] := n(n+1)(n+2)/6;  (*A000292*)
a[1] = 0; h = 128;
c = (u[#1] &) /@ Range[h];
d = (Complement[Range[Max[#1]], #1] &)[c]; (*A145397*)
Table[a[d[[n]]] = 1 - a[n], {n, 1, h - 1}];
Table[a[c[[n]]] = a[n], {n, 1, h}]   (*A189017*)
Flatten[Position[%, 0]]  (*A189018*)
Flatten[Position[%%, 1]] (*A189019*)

Prog

(Python)
from functools import lru_cache
from math import comb
from sympy import integer_nthroot
@lru_cache(maxsize=None)
def A189017(n):
    if n == 1: return 0
    m = integer_nthroot(6*n, 3)[0]
    return A189017(m) if n==(k:=comb(m+2, 3)) else A189017(n-m+(n<=k))^1 # Chai Wah Wu, Nov 07 2025

Crossrefs

Sequence in context: A285196 A189673 A189014 * A189132 A189203 A296028

Adjacent sequences: A189014 A189015 A189016 * A189018 A189019 A189020

Keyword

nonn

Author

Clark Kimberling, Apr 15 2011

Status

approved