A007608 Nonnegative integers in base -4.

0, 1, 2, 3, 130, 131, 132, 133, 120, 121, 122, 123, 110, 111, 112, 113, 100, 101, 102, 103, 230, 231, 232, 233, 220, 221, 222, 223, 210, 211, 212, 213, 200, 201, 202, 203, 330, 331, 332, 333, 320, 321, 322, 323, 310, 311, 312, 313, 300, 301, 302, 303, 13030

Offset: 0

Author

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Keywords

• base
• nice
• easy
• nonn

Comments

The base 2i representation (quater-imaginary representation) of nonnegative integers is obtained by interleaving with zeros, cf. A212494.

More precisely, a(n) is the number n written in base -4; numbers [which represent some nonnegative integer] in base -4 are 0, 1, 2, 3, 100, 101, 102, 103, 110, 111, 112, 113, 120, 121, 122, 123, 130, 131, 132, 133, ... (A212556) - M. F. Hasler, May 20 2012

References

• D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Joerg Arndt, Table of n, a(n) for n = 0..1000
• Matthew Szudzik, A Mathematica programming contest
• Eric Weisstein's World of Mathematics, Negabinary
• Wikipedia, Negative base

Crossrefs

Cf. A212556 (sorted), A066323 (sum of digits), A212526 (negative integers in base -4).

Other bases: A007090, A039724, A073785, A073786, A073787, A073788, A073789, A073790 & A039723.

Programs

Haskell

a007608 0 = 0
a007608 n = a007608 n' * 10 + m where
   (n', m) = if r < 0 then (q + 1, r + 4) else (q, r)
             where (q, r) = quotRem n (negate 4)
-- Reinhard Zumkeller, Jul 15 2012

Mathematica

ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]]; Table[ ToNegaBases[n, 4], {n, 0, 55}]

PARI

A007608(n,s="")={until(!n\=-4,s=Str(n%-4,s));eval(s)}  \\ M. F. Hasler, May 20 2012

Python

def A007608(n):
    s, q = '', n
    while q >= 4 or q < 0:
        q, r = divmod(q, -4)
        if r < 0:
            q += 1
            r += 4
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016