A379184 a(n) = A379183(n) mod 2.

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

Offset

1

Comments

For the interesting properties of this sequence and A379183, see the article "Finding missing gems amidst chaos".

Using Go language code found under Links, a search up to 10^9 terms counts irreducible missing binary words of length L = 1..20 as follows: [0, 0, 0, 0, 0, 0, 0, 0, 0, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]. Can anyone prove the 256 missing words at L=10 are never found in unbounded searching?

A missing word is said to be irreducible if none of its subwords are missing.

Links

Table of n, a(n) for n=1..105.

Brad Klee, Finding missing gems amidst chaos, Wolfram Community, 2024.

Bradley Klee, Conjecture Certificate (golang)

Stephen Wolfram, Nestedly Recursive Functions, 2024.

Maple

b:= proc(n) option remember; `if`(n<4, signum(n-1), n-b(b(n-2))) end:
a:= n-> irem(b(n), 2):
seq(a(n), n=1..105);  # Alois P. Heinz, Dec 21 2024

Mathematica

Block[{a}, a[1]=0; a[2]=a[3]=1; a[n_]:=a[n]=n-a[a[n-2]]; Mod[a/@Range[50], 2]]

Prog

(PARI)
a(n)={my(b); b=vector(n); for(n=1, n, b[n]=if(
n==1, 0, n==2, 1, n==3, 1, n-b[b[n-2]])); b[n]%2}
(Go)
func a(n int) int {
    b := make([]int, n+1);
    copy(b, []int{0, 0, 1, 1});
    for i:=4; i < n+1; i++ {
        b[i] = i-b[b[i-2]];
    };
    return b[n]%2
}

Crossrefs

Cf. A379183, A171587, A379274, A379275.

Sequence in context: A295890 A342704 A284622 * A215581 A156595 A286493

Adjacent sequences: A379181 A379182 A379183 * A379185 A379186 A379187

Keyword

nonn,new

Author

Bradley Klee, Dec 17 2024

Status

approved