A212529 -id:A212529 - cp's OEIS frontend

A039724 a(n) is the negabinary expansion of n, that is, the expansion of n in base -2.

0, 1, 110, 111, 100, 101, 11010, 11011, 11000, 11001, 11110, 11111, 11100, 11101, 10010, 10011, 10000, 10001, 10110, 10111, 10100, 10101, 1101010, 1101011, 1101000, 1101001, 1101110, 1101111, 1101100, 1101101, 1100010, 1100011, 1100000, 1100001, 1100110, 1100111, 1100100

Offset: 0

Robert Lozyniak (11(AT)onna.com)

The numbers written in base -2.

a(A007583(n)) are the only terms with all 1s digits; the number of digits = 2n + 1. - Bob Selcoe, Aug 21 2016

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

M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 101.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

William A. Tedeschi, Table of n, a(n) for n = 0..10000
Joerg Arndt, Matters Computational (The Fxtbook), p. 58-59
Roberto Avanzi, Gerhard Frey, Tanja Lange, and Roger Oyono, On using expansions to the base of -2, International Journal of Computer Mathematics, 81:4 (2004), pp. 403-406. arXiv:math/0312060 [math.NT], 2003.
Jaime Rangel-Mondragon, Negabinary Numbers to Decimal
Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Negabinary
Wikipedia, Negative base

Nonnegative numbers in negative bases: A039723 (b=-10), this sequence (b=-2), A073785 (b=-3), A007608 (b=-4), A073786 (b=-5), A073787 (b=-6), A073788 (b=-7), A073789 (b=-8), A073790 (b=-9).
Cf. A212529 (negative numbers in base -2).
Cf. A005351, A007583.

a039724 0 = 0
a039724 n = a039724 n' * 10 + 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

f:= proc(n) option remember; 10*floor((n mod 4)/2) + (n mod 2) + 100*procname(round(n/4)) end proc:
f(0):= 0:
seq(f(i),i=0..100); # Robert Israel, Feb 24 2016

ToNegaBases[ i_Integer, b_Integer ] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[ (#1 - Mod[ #1, b ])/-b &, i, #1 != 0 & ], b ] ] ] ]; Table[ ToNegaBases[ n, 2 ], {n, 0, 31} ]

A039724(n)=if(n,A039724(n\(-2))*10+bittest(n,0)) \\ M. F. Hasler, Oct 16 2018

def A039724(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]) # Chai Wah Wu, Apr 09 2016

G.f. g(x) satisfies g(x) = (x + 10*x^2 + 11*x^3)/(1 - x^4) + 100(1 + x + x^2 + x^3)*g(x^4)/x^2. - Robert Israel, Feb 24 2016

More terms from Eric W. Weisstein

A005352 Base -2 representation of -n reinterpreted as binary.

Original entry on oeis.org

3, 2, 13, 12, 15, 14, 9, 8, 11, 10, 53, 52, 55, 54, 49, 48, 51, 50, 61, 60, 63, 62, 57, 56, 59, 58, 37, 36, 39, 38, 33, 32, 35, 34, 45, 44, 47, 46, 41, 40, 43, 42, 213, 212, 215, 214, 209, 208, 211, 210, 221, 220, 223, 222, 217, 216, 219, 218, 197, 196, 199, 198, 193, 192, 195, 194, 205, 204, 207, 206, 201, 200

Offset: 1

N. J. A. Sloane

			a(4) = 12 because the negabinary representation of -4 is 1100, and in ordinary binary that is 12.
a(5) = 15 because the negabinary representation of -5 is 1111, and in binary that is 15.

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).

Joerg Arndt, Table of n, a(n) for n = 1..1000
Joerg Arndt, Matters Computational (The Fxtbook), p. 58-59.
Eric Weisstein's World of Mathematics, Negabinary
A. Wilks, Email, May 22 1991

Complement of A005351 in natural numbers.

Cf. A212529.

a005352 = a005351 . negate  -- Reinhard Zumkeller, Feb 05 2014

(* This function comes from the Weisstein page *)
Negabinary[n_Integer] := Module[{t = (2/3)(4^Floor[Log[4, Abs[n] + 1] + 2] - 1)}, IntegerDigits[BitXor[n + t, t], 2]];
Table[FromDigits[Negabinary[n], 2], {n, -1, -50, -1}]
(* Alonso del Arte, Apr 04 2011 *)

a(n) = my(t=(32*4^logint(n+1,4)-2)/3); bitxor(t-n, t); \\ Ruud H.G. van Tol, Oct 19 2023

def A005352(n):
    return ((b:=(4 << (n.bit_length() | 1)) // 3) - n)^b # Adrienne Leonardo, Feb 03 2025

a(n) = A005351(-n). - Reinhard Zumkeller, Feb 05 2014

A305238 Negative numbers in base -10.

Original entry on oeis.org

19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 79, 78, 77, 76, 75

Offset: 1

Jianing Song, Jun 19 2018

a(n) = A039723(-n).
Also base -10 representation of -n reinterpreted as decimal numbers.
The first comment is slightly misleading because sequence A039723 isn't defined for n < 0, and none of the terms a(n) here is a term of A039723. However, it can be seen as the definition of the extension of A039723 to negative indices. Also, the (naïve) recursive definition or implementation of A039723 requires that function to be defined for negative arguments, and using the generic formula it will work as expected for -n, n > 0. - M. F. Hasler, Oct 16 2018

			-1 in base -10 is represented as 19 (1*(-10) + 9 = -1), so a(1) = 19;
-11 in base -10 is represented as 29 (2*(-10) + 9 = -11), so a(11) = 29;
-99 in base -10 is represented as 1901 (1*(-10)^3 + 9*(-10)^2 + 1 = -99), so a(99) = 1901.

Jianing Song, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Negadecimal
Eric Weisstein's World of Mathematics, Negabinary
Wikipedia, Negative base

Cf. A039724 (nonnegative numbers in base -2), A212529 (negative numbers in base -2), A007608 (nonnegative numbers in base -4), A212526 (negative numbers in base -4), A039723 (nonnegative numbers in base -10).

A305238(n)=A039723(-n) \\ M. F. Hasler, Oct 16 2018

A317050 a(0) = 0 and for any n >= 0, a(n+1) is obtained by changing the rightmost possible digit in the negabinary representation of a(n) so as to get a value not yet in the sequence.

Original entry on oeis.org

0, 1, -1, -2, 2, 3, 5, 4, -4, -3, -5, -6, -10, -9, -7, -8, 8, 9, 7, 6, 10, 11, 13, 12, 20, 21, 19, 18, 14, 15, 17, 16, -16, -15, -17, -18, -14, -13, -11, -12, -20, -19, -21, -22, -26, -25, -23, -24, -40, -39, -41, -42, -38, -37, -35, -36, -28, -27, -29, -30

Offset: 0

Rémy Sigrist, Jul 20 2018

Binary Gray code, interpreted as negabinary number.
This sequence is a bijection from nonnegative integers to signed integers.
This sequence has similarities with A317018; in both sequences, the negabinary representations of consecutive terms differ exactly by one digit.

			The first terms, alongside their negabinary representation, are:
  n   a(n)  nega(a(n))
  --  ----  ----------
   0     0        0
   1     1        1
   2    -1       11
   3    -2       10
   4     2      110
   5     3      111
   6     5      101
   7     4      100
   8    -4     1100
   9    -3     1101
  10    -5     1111
  11    -6     1110
  12   -10     1010
  13    -9     1011
  14    -7     1001
  15    -8     1000
  16     8    11000
  17     9    11001
  18     7    11011
  19     6    11010
  20    10    11110
a(8) = -4 because nega(a(7)) = 100. Changing the rightmost digit gives 101 of which the decimal value in the sequence. Similarily, changing to 110 and 000 gives no new term. Changing to 1100 does so a(8) is the decimal value of 1100 which is -4. - _David A. Corneth_, Jul 22 2018

Rémy Sigrist, Table of n, a(n) for n = 0..16383
Eric Weisstein's World of Mathematics, Negabinary.
Wikipedia, Negative base.

Cf. A003188, A039724, A053985, A212529, A317018.

a(n) = fromdigits(binary(bitxor(n, n>>1)), -2)

a(n) = A053985(A003188(n)).

A331892 Positive numbers k such that the negabinary expansion (A039724) of -k is palindromic.

Original entry on oeis.org

1, 5, 7, 17, 21, 31, 35, 57, 65, 85, 93, 119, 127, 147, 155, 201, 217, 257, 273, 325, 341, 381, 397, 455, 471, 511, 527, 579, 595, 635, 651, 745, 777, 857, 889, 993, 1025, 1105, 1137, 1253, 1285, 1365, 1397, 1501, 1533, 1613, 1645, 1767, 1799, 1879, 1911, 2015

Offset: 1

Amiram Eldar, Jan 30 2020

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

			5 is a term since the negabinary representation of -5 is 1111 which is palindromic.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002113, A005352, A006995, A014190, A039724, A094202, A212529, A331191, A331891.

negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; Select[Range[2000], PalindromeQ @ negabin[-#] &]

A317018 Sequence of distinct signed integers such that a(1) = 0 and for any n > 0, the negabinary representation of a(n+1) differ by exactly one digit from the negabinary representation of a(n) and has the smallest possible absolute value (in case of a tie, choose the integer with the rightmost difference).

Original entry on oeis.org

0, 1, -1, -2, 2, 3, 5, -3, -4, 4, 20, 12, 8, 6, 7, 9, -7, -8, -10, -6, -5, -9, -41, 23, 22, 24, 25, 29, 27, 26, 28, 36, -28, -12, -11, -13, -14, -18, 14, 15, 17, -15, -16, 16, 80, 48, 32, 30, 31, 33, -31, -27, -29, -30, -34, -32, -40, -24, -20, -19, 13, 11, 10

Offset: 1

Rémy Sigrist, Jul 19 2018

This sequence has similarities with A316995; in both sequences, the absolute value of the difference of two consecutive terms is a power of 2.

This sequence also has similarities with A163252.

			The first terms, alongside their negabinary representation, are:
  n   a(n)  nega(a(n))
  --  ----  ----------
   1     0        0
   2     1        1
   3    -1       11
   4    -2       10
   5     2      110
   6     3      111
   7     5      101
   8    -3     1101
   9    -4     1100
  10     4      100
  11    20    10100
  12    12    11100
  13     8    11000
  14     6    11010
  15     7    11011
  16     9    11001
  17    -7     1001
  18    -8     1000
  19   -10     1010
  20    -6     1110

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Line plot of the first 10000 terms
Rémy Sigrist, PARI program for A317018
Eric Weisstein's World of Mathematics, Negabinary.
Wikipedia, Negative base.

Cf. A039724, A163252, A212529, A316995.

PARI
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See Links section.
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A320642 Number of 1's in the base-(-2) expansion of -n.

Original entry on oeis.org

2, 1, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 5, 4, 3, 2, 4, 3, 5, 4, 6, 5, 4, 3, 5, 4, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 5, 4, 3, 2, 4, 3, 5, 4, 6, 5, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 6, 5, 4, 3, 5, 4, 3, 2, 4, 3, 5, 4, 6, 5, 4, 3, 5, 4, 6, 5, 7, 6, 5, 4, 6, 5, 7, 6, 8, 7, 6

Offset: 1

Jianing Song, Oct 18 2018

Number of 1's in A212529(n).
Define f(n) as: f(0) = 0, f(-2*n) = f(n), f(-2*n+1) = f(n) + 1, then a(n) = f(-n), n >= 1. See A027615 for the other half of f.
For k > 1, the earliest occurrence of k is n = A086893(k-1).

			A212529(11) = 110101 which has four 1's, so a(11) = 4.
A212529(25) = 111011 which has five 1's, so a(25) = 5.
A212529(51) = 11011101 which has six 1's, so a(51) = 6.

Jianing Song, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Negabinary.
Eric Weisstein's World of Mathematics, Negadecimal.
Wikipedia, Negative base.

Cf. A000120, A005352, A027615, A086893, A212529.

b[n_] := b[n] = b[Quotient[n - 1, -2]] + Mod[n, 2]; b[0] = 0; a[n_] := b[-n]; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)

b(n) = if(n==0, 0, b(n\(-2))+n%2)
a(n) = b(-n)

a(n) == -n (mod 3).

a(n) = A000120(A005352(n)). - Michel Marcus, Oct 23 2018

A331893 Positive numbers k such that both k and -k are a palindromes in negabinary representation.

Original entry on oeis.org

1, 5, 7, 17, 21, 31, 57, 65, 85, 127, 155, 217, 257, 273, 325, 341, 455, 511, 635, 857, 889, 993, 1025, 1105, 1253, 1285, 1365, 1799, 2047, 2159, 2555, 2667, 3417, 3577, 3641, 3937, 4097, 4161, 4369, 4433, 4965, 5125, 5189, 5397, 5461, 6951, 7175, 7967, 8191

Offset: 1

Amiram Eldar, Jan 30 2020

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

			5 is a term since the negabinary representation of 5, 101, and the negabinary representation of -5, 1111, are both palindromic.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A331891 and A331892.

Cf. A005352, A039724, A212529.

negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; nbPalinQ[n_] := And @@ (PalindromeQ @ negabin[#] & /@ {n, -n}); Select[Range[2^13], nbPalinQ]

A320636 Negative numbers in base -3.

Original entry on oeis.org

12, 11, 10, 22, 21, 20, 1202, 1201, 1200, 1212, 1211, 1210, 1222, 1221, 1220, 1102, 1101, 1100, 1112, 1111, 1110, 1122, 1121, 1120, 1002, 1001, 1000, 1012, 1011, 1010, 1022, 1021, 1020, 2202, 2201, 2200, 2212, 2211, 2210, 2222, 2221, 2220, 2102, 2101, 2100, 2112

Offset: 1

Jianing Song, Oct 18 2018

Extend A073785 to negative-indexed terms, then a(n) = A073785(-n).

			-7 in base -3 is represented as 1202 (1*(-3)^3 + 2*(-3)^2 + 2 = -7), so a(7) = 1202;
-16 in base -3 is represented as 1102 (1*(-3)^3 + 1*(-3)^2 + 2 = -16), so a(16) = 1102;
-40 in base -3 is represented as 2222 (2*(-3)^3 + 2*(-3)^2 + 2*(-3) + 2 = -99), so a(40) = 2222.

Eric Weisstein's World of Mathematics, Negadecimal
Eric Weisstein's World of Mathematics, Negabinary
Wikipedia, Negative base

Nonnegative numbers in negative bases: A039723 (b=-10), A039724 (b=-2), A073785 (b=-3), A007608 (b=-4), A073786 (b=-5), A073787 (b=-6), A073788 (b=-7), A073789 (b=-8), A073790 (b=-9).

Negative numbers in negative bases: A305238 (b=-10), A212529 (b=-2), this sequence (b=-3), A212526 (b=-4).

A073785 = base(n, b=-3) = if(n, base(n\b, b)*10 + n%b, 0)
a(n) = A073785(-n)

def A073785(n): # after Reinhard Zumkeller
    if n == 0: return 0
    (q, r) = divmod(n, -3)
    (nn, m) = (q, r) if r >= 0 else (q+1, r+3)
    return A073785(nn)*10 + m
def a(n): return A073785(-n)
print([a(n) for n in range(1, 47)]) # Michael S. Branicky, Dec 11 2021

cp's OEIS Frontend

A039724 a(n) is the negabinary expansion of n, that is, the expansion of n in base -2.

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A005352 Base -2 representation of -n reinterpreted as binary.

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A305238 Negative numbers in base -10.

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A317050 a(0) = 0 and for any n >= 0, a(n+1) is obtained by changing the rightmost possible digit in the negabinary representation of a(n) so as to get a value not yet in the sequence.

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A331892 Positive numbers k such that the negabinary expansion (A039724) of -k is palindromic.

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A320642 Number of 1's in the base-(-2) expansion of -n.

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