A073788 -id:A073788 - 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

A039723 Numbers in base -10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 150, 151, 152, 153, 154, 155

Offset: 0

Robert Lozyniak (11(AT)onna.com)

			Decimal 25 is "185" in base -10 because 100 - 80 + 5 = 25.

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

Nonnegative numbers in negative bases: this sequence (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).
Cf. A051022.
Cf. A305238: negative numbers in base -10.

a039723 0 = 0
a039723 n = a039723 n' * 10 + m where
   (n',m) = if r < 0 then (q + 1, r + 10) else qr where
            qr@(q, r) = quotRem n (negate 10)
-- Reinhard Zumkeller, Apr 20 2011

ToNegaBases[i_Integer, b_Integer] := FromDigits@ Rest@ Reverse@ Mod[ NestWhileList[(# - Mod[ #, b])/-b &, i, # != 0 &], b]

A039723 = base(n, b=-10) = if(n, base(n\b, b)*10 + n%b, 0) \\ M. F. Hasler, Oct 16 2018 [Corrected by Jianing Song, Oct 21 2018]

def A039723(n):
    s, q = '', n
    while q >= 10 or q < 0:
        q, r = divmod(q, -10)
        if r < 0:
            q += 1
            r += 10
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 10 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

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

A073794 Replace 7^k with (-7)^k in base 7 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, -7, -6, -5, -4, -3, -2, -1, -14, -13, -12, -11, -10, -9, -8, -21, -20, -19, -18, -17, -16, -15, -28, -27, -26, -25, -24, -23, -22, -35, -34, -33, -32, -31, -30, -29, -42, -41, -40, -39, -38, -37, -36, 49, 50, 51, 52, 53, 54, 55, 42, 43, 44, 45, 46, 47, 48, 35, 36, 37, 38, 39, 40, 41

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 7 representation for n (in lexicographic order) converted from base -7 to base 10.

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. A007093, A073788, A053985, A065369, A073791, A073792, A073793, A073795, A073796 & A073835.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 7]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 7]], {n, 1, 80}]; b

a(7*k+m) = -7*a(k)+m for 0 <= m < 7. - Chai Wah Wu, Jan 16 2020

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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A007608 Nonnegative integers in base -4.

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A039723 Numbers in base -10.

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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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A073789 Numbers in base -8.

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