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A005351
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Base -2 representation for n regarded as base 2, then evaluated.
(Formerly M4059)
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14
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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EXAMPLE
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2 = 4+(-2)+0 = 110 => 6, 3 = 4+(-2)+1 = 111 => 7, ..., 6 = (16)+(-8)+0+(-2)+0 = 11010 => 26.
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MATHEMATICA
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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 *)
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PROG
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(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)
(Python)
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
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CROSSREFS
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Cf. A185269 (primes in this sequence).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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