A073785 Numbers in base -3.

0, 1, 2, 120, 121, 122, 110, 111, 112, 100, 101, 102, 220, 221, 222, 210, 211, 212, 200, 201, 202, 12020, 12021, 12022, 12010, 12011, 12012, 12000, 12001, 12002, 12120, 12121, 12122, 12110, 12111, 12112, 12100, 12101, 12102, 12220, 12221, 12222

Offset: 0

Author

Robert G. Wilson v, Aug 11 2002

Keywords

• base
• easy
• nonn

References

• D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Jaime Rangel-Mondragon, Negabinary Numbers to Decimal
• Eric Weisstein's World of Mathematics, Negabinary
• Wikipedia, Negative base

Crossrefs

Cf. A007089.

Nonnegative numbers in negative bases: A039723 (b=-10), A039724 (b=-2), this sequence (b=-3), A007608 (b=-4), A073786 (b=-5), A073787 (b=-6), A073788 (b=-7), A073789 (b=-8), A073790 (b=-9).

Cf. A320636 (negative numbers in base -3).

Programs

Haskell

a073785 0 = 0
a073785 n = a073785 n' * 10 + m where
   (n', m) = if r < 0 then (q + 1, r + 3) else (q, r)
             where (q, r) = quotRem n (negate 3)
-- Reinhard Zumkeller, Jul 07 2012

Mathematica

ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]]; Table[ ToNegaBases[n, 3], {n, 0, 45}]

PARI

A073785 = base(n, b=-3) = if(n, base(n\b, b)*10 + n%b, 0) \\ Jianing Song, Oct 20 2018

Python

def A073785(n):
    s, q = '', n
    while q >= 3 or q < 0:
        q, r = divmod(q, -3)
        if r < 0:
            q += 1
            r += 3
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016