A212526 -id:A212526 - cp's OEIS frontend

A007608 Nonnegative integers in base -4.

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

A212529 Negative numbers in base -2.

Original entry on oeis.org

11, 10, 1101, 1100, 1111, 1110, 1001, 1000, 1011, 1010, 110101, 110100, 110111, 110110, 110001, 110000, 110011, 110010, 111101, 111100, 111111, 111110, 111001, 111000, 111011, 111010, 100101, 100100, 100111, 100110, 100001, 100000, 100011, 100010, 101101, 101100, 101111, 101110, 101001, 101000, 101011, 101010, 11010101

Offset: 1

Joerg Arndt, May 20 2012

The formula a(n) = A039724(-n) is slightly misleading because sequence A039724 isn't defined for n < 0, and none of the terms a(n) is a term of A039724. It can be seen as the definition of the extension of A039724 to negative indices. Also, recursive definitions or implementations of A039724 require that function to be defined for negative arguments, and using a generic formula it will work as expected for -n, n > 0. - M. F. Hasler, Oct 18 2018

Joerg Arndt, Table of n, a(n) for n = 1..1000
Joerg Arndt, Matters Computational (The Fxtbook), pp. 58-59.

Cf. A039724 (nonnegative numbers in base -2).
Cf. A007608 (nonnegative numbers in base -4), A212526 (negative numbers in base -4).
Cf. A005352.

a212529 = a039724 . negate  -- Reinhard Zumkeller, Feb 05 2014

a:= proc(n) local d, i, l, m;
      m:= n; l:= NULL;
      for i from 0 while m>0 do
        d:= irem(m, 2, 'm');
        if d=1 and irem(i, 2)=0 then m:= m+1 fi;
        l:= d, l
      od; parse(cat(l))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 20 2012

negabin[n_] := negabin[n] = If[n == 0, 0, negabin[Quotient[n - 1, -2]]*10 + Mod[n, 2]]; a[n_] := negabin[-n]; Array[a, 50] (* Amiram Eldar, Jul 23 2023 *)

A212529(n)=A039724(-n) \\ M. F. Hasler, Oct 16 2018

def A212529(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

a(n) = A039724(-n). - Reinhard Zumkeller, Feb 05 2014

A256441 Binary representation of base-(i-1) expansion of -n: replace i-1 with 2 in base-(i-1) expansion of -n.

Original entry on oeis.org

0, 29, 28, 17, 16, 205, 204, 193, 192, 221, 220, 209, 208, 7437, 7436, 7425, 7424, 7453, 7452, 7441, 7440, 7629, 7628, 7617, 7616, 7645, 7644, 7633, 7632, 7181, 7180, 7169, 7168, 7197, 7196, 7185, 7184, 7373, 7372, 7361, 7360, 7389, 7388, 7377, 7376, 4365

Offset: 0

Paul Tek, Mar 29 2015

Here i = sqrt(-1).
From Jianing Song, Jan 22 2023: (Start)
Also binary representation of base-(-1-i) expansion of -n.
Write out -n in base -4 (A212526), change each digit 0, 1, 2, 3 to 0000, 0001, 1100, 1101 respectively, then interpret as a binary number. (End)

			a(5) = 205 = 2^7 + 2^6 + 2^3 + 2^2 + 2^0 since (i-1)^7 + (i-1)^6 + (i-1)^3 + (i-1)^2 + (i-1)^0 = -5.

Paul Tek, Table of n, a(n) for n = 0..10000
Paul Tek, Perl program for this sequence
Wikipedia, Complex-base system

Cf. A066321.

a(n) = my(v = [-n,0], x=0, digit=0, a, b); while(v!=[0,0], a=v[1]; b=v[2]; v[1]=-2*(a\2)+b; v[2]=-(a\2); x+=(a%2)*2^digit; digit++); x \\ Jianing Song, Jan 22 2023; [a,b] represents the number a + b*(-1+i)

Perl
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For n >= 1, a(4*n-0..3) = 16 * A066321(n) + 0, 1, 12, 13 respectively. - Jianing Song, Jan 22 2023

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

A212542 Base 2i representation of negative integers.

Original entry on oeis.org

103, 102, 101, 100, 203, 202, 201, 200, 303, 302, 301, 300, 1030003, 1030002, 1030001, 1030000, 1030103, 1030102, 1030101, 1030100, 1030203, 1030202, 1030201, 1030200, 1030303, 1030302, 1030301, 1030300, 1020003, 1020002, 1020001, 1020000, 1020103, 1020102, 1020101, 1020100, 1020203, 1020202, 1020201, 1020200, 1020303

Offset: 1

Joerg Arndt, May 20 2012

Omitting digits for odd powers of 2i (all 0's for the imaginary parts) (e.g. 1030003 --> 1303) gives A212526 (negative integers in base -4).

			a(13) = 1030003: 1*(2*i)^6 + 0 + 3*(2*i)^4 + 0 + 0 + 0 + 3*(2*i)^0 = -64 + 48 + 3 = -13.

Joerg Arndt, Table of n, a(n) for n = 1..1000

Cf. A212494 (base 2i representation of nonnegative integers).

a:= proc(n) local d, i, l, m;
      m:= n; l:= NULL;
      for i from 0 while m>0 do
        d:= irem(m, 4, 'm');
        if irem(i, 2)=0 and d>0 then d:= 4-d; m:= m+1 fi;
        l:= d, 0, l
      od; parse(cat(l))/10
    end:
seq(a(n), n=1..60); # Alois P. Heinz, May 20 2012

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

A360034 Binary representation of -n in base i-1.

Original entry on oeis.org

0, 11101, 11100, 10001, 10000, 11001101, 11001100, 11000001, 11000000, 11011101, 11011100, 11010001, 11010000, 1110100001101, 1110100001100, 1110100000001, 1110100000000, 1110100011101, 1110100011100, 1110100010001, 1110100010000, 1110111001101, 1110111001100, 1110111000001

Offset: 0

Jianing Song, Jan 22 2023

Note that each Gaussian integer has one and only one base-(i-1) representation.
Also binary representation of -n in base -1-i.
Write out -n in base -4 (A212526), then change each digit 0, 1, 2, 3 to 0000, 0001, 1100, 1101 respectively.

			a(1) = 11101 since -1 = (i-1)^4 + (i-1)^3 + (i-1)^2 + (i-1)^0. Also, the base-(-4) representation of -1 is 13_(-4), so changing 1 to 0001 and 3 to 1101 yields 11101.
a(5) = 11001101 since -5 = (i-1)^7 + (i-1)^6 + (i-1)^3 + (i-1)^2 + (i-1)^0. Also, the base-(-4) representation of -5 is 23_(-4), so changing 2 to 1100 and 3 to 1101 yields 11001101.

Jianing Song, Table of n, a(n) for n = 0..16384
Wikipedia, Complex-base system

This is A256441 converted from base 10 to base 2. Cf. also A271472.

a(n) = my(v = [-n,0], x=0, digit=0, a, b); while(v!=[0,0], a=v[1]; b=v[2]; v[1]=-2*(a\2)+b; v[2]=-(a\2); x+=(a%2)*10^digit; digit++); x \\ Jianing Song, Jan 22 2023; [a,b] represents the number a + b*(-1+i)

For n >= 1, a(4*n-0..3) = 10000 * A271472(n) + 0, 1, 1100, 1101 respectively.

cp's OEIS Frontend

A007608 Nonnegative integers in base -4.

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A212529 Negative numbers in base -2.

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A256441 Binary representation of base-(i-1) expansion of -n: replace i-1 with 2 in base-(i-1) expansion of -n.

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A305238 Negative numbers in base -10.

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A212542 Base 2i representation of negative integers.

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A320636 Negative numbers in base -3.

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A360034 Binary representation of -n in base i-1.

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