A212529 Negative numbers in base -2.

11, 10, 1101, 1100, 1111, 1110, 1001, 1000, 1011, 1010, 110101, 110100, 110111, 110110, 110001, 110000, 110011, 110010, 111101, 111100, 111111, 111110, 111001, 111000, 111011, 111010, 100101, 100100, 100111, 100110, 100001, 100000, 100011, 100010, 101101, 101100, 101111, 101110, 101001, 101000, 101011, 101010, 11010101

Offset: 1

Author

Joerg Arndt, May 20 2012

Keywords

• nonn
• base

Comments

The formula a(n) = A039724(-n) is slightly misleading because sequence A039724 isn't defined for n < 0, and none of the terms a(n) is a term of A039724. It can be seen as the definition of the extension of A039724 to negative indices. Also, recursive definitions or implementations of A039724 require that function to be defined for negative arguments, and using a generic formula it will work as expected for -n, n > 0. - M. F. Hasler, Oct 18 2018

Links

• Joerg Arndt, Table of n, a(n) for n = 1..1000
• Joerg Arndt, Matters Computational (The Fxtbook), pp. 58-59.

Crossrefs

Cf. A039724 (nonnegative numbers in base -2).

Cf. A007608 (nonnegative numbers in base -4), A212526 (negative numbers in base -4).

Cf. A005352.

Programs

Haskell

a212529 = a039724 . negate  -- Reinhard Zumkeller, Feb 05 2014

Maple

a:= proc(n) local d, i, l, m;
      m:= n; l:= NULL;
      for i from 0 while m>0 do
        d:= irem(m, 2, 'm');
        if d=1 and irem(i, 2)=0 then m:= m+1 fi;
        l:= d, l
      od; parse(cat(l))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 20 2012

Mathematica

negabin[n_] := negabin[n] = If[n == 0, 0, negabin[Quotient[n - 1, -2]]*10 + Mod[n, 2]]; a[n_] := negabin[-n]; Array[a, 50] (* Amiram Eldar, Jul 23 2023 *)

PARI

A212529(n)=A039724(-n) \\ M. F. Hasler, Oct 16 2018

Python

def A212529(n):
    s, q = '', -n
    while q >= 2 or q < 0:
        q, r = divmod(q, -2)
        if r < 0:
            q += 1
            r += 2
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

Formula

a(n) = A039724(-n). - Reinhard Zumkeller, Feb 05 2014