A189014 Zero-one sequence based on pentagonal numbers: a(A000326(k))=a(k); a(A183217(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, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1

Offset

1

Links

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

Formula

Let u=A000217 and v=A014132, so that u(n)=n(3n-1)/2 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(3n-1)/2;  (*A000326*)
a[1] = 0; h = 128;
c = (u[#1] &) /@ Range[h];
d = (Complement[Range[Max[#1]], #1] &)[c]; (*A183217*)
Table[a[d[[n]]] = 1 - a[n], {n, 1, h - 1}];
Table[a[c[[n]]] = a[n], {n, 1, h}]   (*A189014*)
Flatten[Position[%, 0]]  (*A189015*)
Flatten[Position[%%, 1]] (*A189016*)

Prog

(Python)
from functools import lru_cache
from math import isqrt
@lru_cache(maxsize=None)
def A189014(n): return 0 if n==1 else A189014(m) if n==(k:=(m:=isqrt((n<<3)//3)+1>>1)*(3*m-1)>>1) else A189014(n-m+(n<=k))^1 # Chai Wah Wu, Nov 08 2025

Crossrefs

Cf. A188967, A189015, A189016.

Sequence in context: A372258 A285196 A189673 * A189017 A189132 A189203

Adjacent sequences: A189011 A189012 A189013 * A189015 A189016 A189017

Keyword

nonn

Author

Clark Kimberling, Apr 15 2011

Status

approved