A039723 -id:A039723 - 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

A007608 Nonnegative integers in base -4.

Original entry on oeis.org

0, 1, 2, 3, 130, 131, 132, 133, 120, 121, 122, 123, 110, 111, 112, 113, 100, 101, 102, 103, 230, 231, 232, 233, 220, 221, 222, 223, 210, 211, 212, 213, 200, 201, 202, 203, 330, 331, 332, 333, 320, 321, 322, 323, 310, 311, 312, 313, 300, 301, 302, 303, 13030

Offset: 0

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

The base 2i representation (quater-imaginary representation) of nonnegative integers is obtained by interleaving with zeros, cf. A212494.

More precisely, a(n) is the number n written in base -4; numbers [which represent some nonnegative integer] in base -4 are 0, 1, 2, 3, 100, 101, 102, 103, 110, 111, 112, 113, 120, 121, 122, 123, 130, 131, 132, 133, ... (A212556) - M. F. Hasler, May 20 2012

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.
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 = 0..1000
Matthew Szudzik, A Mathematica programming contest
Eric Weisstein's World of Mathematics, Negabinary
Wikipedia, Negative base

Cf. A212556 (sorted), A066323 (sum of digits), A212526 (negative integers in base -4).

Other bases: A007090, A039724, A073785, A073786, A073787, A073788, A073789, A073790 & A039723.

a007608 0 = 0
a007608 n = a007608 n' * 10 + m where
   (n', m) = if r < 0 then (q + 1, r + 4) else (q, r)
             where (q, r) = quotRem n (negate 4)
-- Reinhard Zumkeller, Jul 15 2012

ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]]; Table[ ToNegaBases[n, 4], {n, 0, 55}]

A007608(n,s="")={until(!n\=-4,s=Str(n%-4,s));eval(s)}  \\ M. F. Hasler, May 20 2012

def A007608(n):
    s, q = '', n
    while q >= 4 or q < 0:
        q, r = divmod(q, -4)
        if r < 0:
            q += 1
            r += 4
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A073785 Numbers in base -3.

Original entry on oeis.org

0, 1, 2, 120, 121, 122, 110, 111, 112, 100, 101, 102, 220, 221, 222, 210, 211, 212, 200, 201, 202, 12020, 12021, 12022, 12010, 12011, 12012, 12000, 12001, 12002, 12120, 12121, 12122, 12110, 12111, 12112, 12100, 12101, 12102, 12220, 12221, 12222

Offset: 0

Robert G. Wilson v, Aug 11 2002

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jaime Rangel-Mondragon, Negabinary Numbers to Decimal
Eric Weisstein's World of Mathematics, Negabinary
Wikipedia, Negative base

Cf. A007089.
Nonnegative numbers in negative bases: A039723 (b=-10), A039724 (b=-2), this sequence (b=-3), A007608 (b=-4), A073786 (b=-5), A073787 (b=-6), A073788 (b=-7), A073789 (b=-8), A073790 (b=-9).
Cf. A320636 (negative numbers in base -3).

a073785 0 = 0
a073785 n = a073785 n' * 10 + m where
   (n', m) = if r < 0 then (q + 1, r + 3) else (q, r)
             where (q, r) = quotRem n (negate 3)
-- Reinhard Zumkeller, Jul 07 2012

ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]]; Table[ ToNegaBases[n, 3], {n, 0, 45}]

A073785 = base(n, b=-3) = if(n, base(n\b, b)*10 + n%b, 0) \\ Jianing Song, Oct 20 2018

def A073785(n):
    s, q = '', n
    while q >= 3 or q < 0:
        q, r = divmod(q, -3)
        if r < 0:
            q += 1
            r += 3
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A073786 Numbers in base -5.

Original entry on oeis.org

0, 1, 2, 3, 4, 140, 141, 142, 143, 144, 130, 131, 132, 133, 134, 120, 121, 122, 123, 124, 110, 111, 112, 113, 114, 100, 101, 102, 103, 104, 240, 241, 242, 243, 244, 230, 231, 232, 233, 234, 220, 221, 222, 223, 224, 210, 211, 212, 213, 214, 200, 201, 202, 203

Offset: 0

Robert G. Wilson v, Aug 11 2002

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Negabinary
Prepared and presented by Matthew Szudzik of Wolfram Research, A Mathematica programming contest

Cf. A007091, A039724, A073785, A007608, A073787, A073788, A073789, A073790 & A039723.

ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]]; Table[ ToNegaBases[n, 5], {n, 0, 55}]

def A073786(n):
    s, q = '', n
    while q >= 5 or q < 0:
        q, r = divmod(q, -5)
        if r < 0:
            q += 1
            r += 5
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A073787 Numbers in base -6.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 150, 151, 152, 153, 154, 155, 140, 141, 142, 143, 144, 145, 130, 131, 132, 133, 134, 135, 120, 121, 122, 123, 124, 125, 110, 111, 112, 113, 114, 115, 100, 101, 102, 103, 104, 105, 250, 251, 252, 253, 254, 255, 240, 241, 242, 243, 244, 245

Offset: 0

Robert G. Wilson v, Aug 11 2002

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Negabinary
Prepared and presented by Matthew Szudzik of Wolfram Research, A Mathematica programming contest

Cf. A007092, A039724, A073785, A007608, A073786, A073788, A073789, A073790 & A039723.

ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]]; Table[ ToNegaBases[n, 6], {n, 0, 60}]

def A073787(n):
    s, q = '', n
    while q >= 6 or q < 0:
        q, r = divmod(q, -6)
        if r < 0:
            q += 1
            r += 6
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A073788 Numbers in base -7.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 160, 161, 162, 163, 164, 165, 166, 150, 151, 152, 153, 154, 155, 156, 140, 141, 142, 143, 144, 145, 146, 130, 131, 132, 133, 134, 135, 136, 120, 121, 122, 123, 124, 125, 126, 110, 111, 112, 113, 114, 115, 116, 100, 101, 102, 103, 104, 105

Offset: 0

Robert G. Wilson v, Aug 11 2002

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Negabinary
Prepared and presented by Matthew Szudzik of Wolfram Research, A Mathematica programming contest

Cf. A007093, A039724, A073785, A007608, A073786, A073787, A073789, A073790 & A039723.

ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]]; Table[ ToNegaBases[n, 7], {n, 0, 60}]

def A073788(n):
    s, q = '', n
    while q >= 7 or q < 0:
        q, r = divmod(q, -7)
        if r < 0:
            q += 1
            r += 7
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A073789 Numbers in base -8.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 170, 171, 172, 173, 174, 175, 176, 177, 160, 161, 162, 163, 164, 165, 166, 167, 150, 151, 152, 153, 154, 155, 156, 157, 140, 141, 142, 143, 144, 145, 146, 147, 130, 131, 132, 133, 134, 135, 136, 137, 120, 121, 122, 123, 124, 125, 126

Offset: 0

Robert G. Wilson v, Aug 11 2002

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Prepared and presented by Matthew Szudzik of Wolfram Research, A Mathematica programming contest
Eric Weisstein's World of Mathematics, Negabinary
Wikipedia, Negative base

Cf. A007094, A039724, A073785, A007608, A073786, A073787, A073788, A073790 & A039723.

a073789 0 = 0
a073789 n = a073789 n' * 10 + m where
   (n', m) = if r < 0 then (q + 1, r + 8) else (q, r)
             where (q, r) = quotRem n (negate 8)
-- Reinhard Zumkeller, Jul 07 2012

ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]]; Table[ ToNegaBases[n, 8], {n, 0, 60}]

def A073789(n):
    s, q = '', n
    while q >= 8 or q < 0:
        q, r = divmod(q, -8)
        if r < 0:
            q += 1
            r += 8
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A073790 Numbers in base -9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 180, 181, 182, 183, 184, 185, 186, 187, 188, 170, 171, 172, 173, 174, 175, 176, 177, 178, 160, 161, 162, 163, 164, 165, 166, 167, 168, 150, 151, 152, 153, 154, 155, 156, 157, 158, 140, 141, 142, 143, 144, 145, 146, 147, 148, 130, 131

Offset: 0

Robert G. Wilson v, Aug 11 2002

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Negabinary
Prepared and presented by Matthew Szudzik of Wolfram Research, A Mathematica programming contest

Cf. A007095, A039724, A073785, A007608, A073786, A073787, A073788, A073789 & A039723.

ToNegaBases[i_Integer, b_Integer] := FromDigits@ Rest@ Reverse@ Mod[ NestWhileList[(# - Mod[ #, b])/-b &, i, # != 0 &], b]; Table[ ToNegaBases[n, 9], {n, 0, 60}]

def A073790(n):
    s, q = '', n
    while q >= 9 or q < 0:
        q, r = divmod(q, -9)
        if r < 0:
            q += 1
            r += 9
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A073835 Replace 10^k with (-10)^k in decimal expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -30, -29, -28, -27, -26, -25, -24, -23, -22, -21, -40, -39, -38, -37, -36, -35, -34, -33, -32, -31, -50, -49, -48, -47, -46, -45, -44, -43, -42, -41, -60

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 10 representation for n (in lexicographic order) converted from base -10 to base 10.

A bijection from N = [0..oo) to Z = (-oo..+oo), or enumeration of the integers. - M. F. Hasler, Oct 17 2018

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020

Cf. A001477, A039723, A053985, A065369, A073791, A073792, A073793, A073794, A073795 & A073796.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[FromDigits[ IntegerDigits[n, 10]], {n, 0, 80}]; b = {}; Do[ b = Append[b, f[a[[n]], 10]], {n, 1, 80}]; b (* Typo fixed by Harvey P. Dale, Oct 03 2013 *)

a(n)=fromdigits(digits(n),-10) \\ M. F. Hasler, Oct 17 2018

a(10*k+m) = -10*a(k)+m for 0 <= m < 10. - Chai Wah Wu, Jan 16 2020

A051022 Interpolate 0's between each pair of digits of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 500, 501, 502, 503, 504, 505

Offset: 0

Eric W. Weisstein, Dec 11 1999

These numbers have the same decimal and negadecimal representations.

Or fixed points of decimal negadecimal conversion. - Gerald Hillier, Apr 23 2015

			a(23) = 203.
a(99) = 909.
a(100) = 10000.
a(101) = 10001.
a(111) = 10101.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Negadecimal

Cf. A039723, A063010, A092908 (primes), A092909 (on primes), A338754 (*11).

In other bases: A000695, A037314, A276089.

a051022 n = if n < 10 then n else a051022 n' * 100 + r
            where (n', r) = divMod n 10
-- Reinhard Zumkeller, Apr 20 2011
(HP 49G calculator)
« "" + SREV 0 9
  FOR i i "" + DUP 0 + SREPL DROP
  NEXT SREV OBJ->
». Gerald Hillier, Apr 23 2015

M:= 3: # to get a(0) to a(10^M-1)
A:= 0:
for d from 1 to M do
  A:= seq(seq(a*100+b,b=0..9),a=A);
od:
A; # Robert Israel, Apr 23 2015

Table[FromDigits[Riffle[IntegerDigits[n],0]],{n,0,60}] (* Harvey P. Dale, Nov 17 2013 *)
ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]];
k = 0; lst = {}; While[k < 1001, If[k == ToNegaBases[k, 10], AppendTo[ lst, k]]; k++]; lst (* Robert G. Wilson v, Jun 11 2014 *)

a(n) = fromdigits(digits(n),100); \\ Kevin Ryde, Nov 07 2020

def a(n): return int("0".join(str(n)))
print([a(n) for n in range(56)]) # Michael S. Branicky, Aug 15 2022

Sums a_i*100^e_i with 0 <= a_i < 10.
a(n) = n if n < 10, otherwise a(floor(n/10))*100 + n mod 10. - Reinhard Zumkeller, Apr 20 2011 [Corrected by Kevin Ryde, Nov 07 2020]
a(n) = A338754(n)/11. - Kritsada Moomuang, Oct 20 2019 [Corrected by Kevin Ryde, Nov 07 2020]

More terms and more precise definition from Jorge Coveiro, Apr 15 2004 and David Wasserman, Feb 26 2008
Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar
Offset fixed by Reinhard Zumkeller, Apr 20 2012

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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A007608 Nonnegative integers in base -4.

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A073785 Numbers in base -3.

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A073786 Numbers in base -5.

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A073787 Numbers in base -6.

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A073788 Numbers in base -7.

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A073789 Numbers in base -8.

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