A005351 - OEIS

%I M4059 #73 Oct 18 2023 17:25:53

%S 0,1,6,7,4,5,26,27,24,25,30,31,28,29,18,19,16,17,22,23,20,21,106,107,

%T 104,105,110,111,108,109,98,99,96,97,102,103,100,101,122,123,120,121,

%U 126,127,124,125,114,115,112,113,118,119,116,117,74,75,72,73,78,79,76

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

%C 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

%C a(n) = n if n is a sum of distinct powers of 4. - _Michael Somos_, Aug 27 2012

%C 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

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

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

%H Reinhard Zumkeller, <a href="/A005351/b005351.txt">Table of n, a(n) for n = 0..8191</a>

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, p. 58-59

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Negabinary.html">Negabinary</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Negative_base">Negative base</a>

%H A. Wilks, <a href="/A005351/a005351.pdf">Email, May 22 1991</a>

%F 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

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

%t 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 *)

%o (Haskell)

%o a005351 0 = 0

%o a005351 n = a005351 n' * 2 + m where

%o    (n', m) = if r < 0 then (q + 1, r + 2) else (q, r)

%o              where (q, r) = quotRem n (negate 2)

%o -- _Reinhard Zumkeller_, Jul 07 2012

%o (Python)

%o def A005351(n):

%o     s, q = '', n

%o     while q >= 2 or q < 0:

%o         q, r = divmod(q, -2)

%o         if r < 0:

%o             q += 1

%o             r += 2

%o         s += str(r)

%o     return int(str(q)+s[::-1],2) # _Chai Wah Wu_, Apr 10 2016

%o (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

%Y Cf. A039724. Complement of A005352.

%Y Cf. A185269 (primes in this sequence).

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

%K nonn,base,easy,nice,look

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Robert G. Wilson v_, Jan 24 2005