A005351 Base -2 representation for n regarded as base 2, then evaluated. (Formerly M4059)

0, 1, 6, 7, 4, 5, 26, 27, 24, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 23, 20, 21, 106, 107, 104, 105, 110, 111, 108, 109, 98, 99, 96, 97, 102, 103, 100, 101, 122, 123, 120, 121, 126, 127, 124, 125, 114, 115, 112, 113, 118, 119, 116, 117, 74, 75, 72, 73, 78, 79, 76

Offset

0,3

Comments

a(n) = n when n is a power of 4. This is because the even-indexed powers of 2 are the same as the even-indexed powers of -2. - Alonso del Arte, Feb 09 2012

a(n) = n if n is a sum of distinct powers of 4. - Michael Somos, Aug 27 2012

Write n = Sum_{i in b(n)} (-2)^(i - 1), which uniquely determines the set of positive integers b(n). Then a(n) = Sum_{i in b(n)} 2^(i - 1). For example, a(7) = 27 because 7 = (-2)^0 + (-2)^1 + (-2)^3 + (-2)^4 and 27 = 2^0 + 2^1 + 2^3 + 2^4. - Gus Wiseman, Jul 26 2019

References

M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 101.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191

Joerg Arndt, Matters Computational (The Fxtbook), p. 58-59

Eric Weisstein's World of Mathematics, Negabinary

Wikipedia, Negative base

A. Wilks, Email, May 22 1991

Formula

a(4n+2) = 4a(n+1)+2, a(4n+3) = 4a(n+1)+3, a(4n+4) = 4a(n+1), a(4n+5) = 4a(n+1)+1, n>-2, a(1)=1. - Ralf Stephan, Apr 06 2004

Example

2 = 4+(-2)+0 = 110 => 6, 3 = 4+(-2)+1 = 111 => 7, ..., 6 = (16)+(-8)+0+(-2)+0 = 11010 => 26.

Mathematica

f[n_] := Module[{t = 2(4^Floor[ Log[4, Abs[n] + 1] + 2] - 1)/3}, BitXor[n + t, t]]; Table[ f[n]], {n, 0, 60}] (* Robert G. Wilson v, Jan 24 2005 *)

Prog

(Haskell)
a005351 0 = 0
a005351 n = a005351 n' * 2 + m where
   (n', m) = if r < 0 then (q + 1, r + 2) else (q, r)
             where (q, r) = quotRem n (negate 2)
-- Reinhard Zumkeller, Jul 07 2012
(Python)
def A005351(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], 2) # Chai Wah Wu, Apr 10 2016
(PARI) a(n) = my(t=(32*4^logint(abs(n)+1, 4)-2)/3); bitxor(n+t, t); \\ Ruud H.G. van Tol, Oct 18 2023

Crossrefs

Cf. A039724. Complement of A005352.

Cf. A185269 (primes in this sequence).

Cf. A000120, A029931, A035327, A053985, A065359, A070939, A326032.

Sequence in context: A019932 A004447 A258989 * A098882 A254374 A019616

Adjacent sequences: A005348 A005349 A005350 * A005352 A005353 A005354

Keyword

nonn,base,easy,nice,look

Author

N. J. A. Sloane

Extensions

More terms from Robert G. Wilson v, Jan 24 2005

Status

approved