A005351 Base -2 representation for n regarded as base 2, then evaluated.

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

Author

N. J. A. Sloane

Keywords

• nonn
• base
• easy
• nice
• look

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

Examples

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

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

Crossrefs

Cf. A039724. Complement of A005352.

Cf. A185269 (primes in this sequence).

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

Programs

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

Mathematica

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

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

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

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

Extensions

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