A256441 -id:A256441 - cp's OEIS frontend

A066321 Binary representation of base-(i-1) expansion of n: replace i-1 with 2 in base-(i-1) expansion of n.

0, 1, 12, 13, 464, 465, 476, 477, 448, 449, 460, 461, 272, 273, 284, 285, 256, 257, 268, 269, 3280, 3281, 3292, 3293, 3264, 3265, 3276, 3277, 3088, 3089, 3100, 3101, 3072, 3073, 3084, 3085, 3536, 3537, 3548, 3549, 3520, 3521, 3532, 3533, 3344, 3345, 3356

Offset: 0

Marc LeBrun, Dec 14 2001

Here i = sqrt(-1).
First differences follow a strange period-16 pattern: 1 11 1 XXX 1 11 1 -29 1 11 1 -189 1 11 1 -29 where XXX is given by A066322. Number of one-bits is A066323.
From Andrey Zabolotskiy, Feb 06 2017: (Start)
(Observations.)
Actually, the sequence of the first differences can be split into blocks of size of any power of 2, and there will be only one position in the block that does not repeat. In this sense, one may say that the first differences follow (almost-)period-2^s pattern for any s > 0.
Specifically, the first differences are given by the formula: a(n+1)-a(n) = A282137(A007814((n xor ...110011001100) + 1)). Here binary representation of n is bitwise-xored with the period-4 bit sequence (A021913 written right-to-left) which is infinite or simply long enough; A007814(m) does not depend on the bits of m other than the least significant 1.
A282137 gives all first differences in the order of decreasing occurrence frequency.
(End)
Penney shows that since (i-1)^4 = -4, the representation a(n) of a real integer n is found by writing n in base -4 using digits 0 to 3 (A007608), changing those digits to bit strings 0000, 0001, 1100, 1101 respectively, and interpreting as binary. - Kevin Ryde, Sep 07 2019

			a(4) = 464 = 2^8 + 2^7 + 2^6 + 2^4 since (i-1)^8 + (i-1)^7 + (i-1)^6 + (i-1)^4 = 4.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 172. (See also exercise 16, p. 177; answer, p. 494.)

Paul Tek, Table of n, a(n) for n = 0..10000
Joerg Arndt, The fxt demos: bit wizardry, radix(-1+i), C++ radix-m1pi.h function bin_real_to_radm1pi().
Solomon I. Khmelnik, Specialized Digital Computer for Operations with Complex Numbers, Questions of Radio Electronics, 12 (1964), 60-82 [in Russian].
W. J. Penney, A "binary" system for complex numbers, JACM 12 (1965), 247-248.
N. J. A. Sloane, Table of n, (I-1)^n for n = 0..100
Paul Tek, Perl program for this sequence
Andrey Zabolotskiy, negation & addition of Gaussian integers written in base i-1 [Python script].

Cf. A066322, A066323, A282137.
See A271472 for the conversion of these decimal numbers to binary.
See A009116 and A009545 for real and imaginary parts of (i-1)^n (except for signs).
See A256441 for expansions of -n.

f:= proc(n) option remember; local t,m;
   t:= n mod 4;
   procname(t) + 16*procname((t-n)/4)
end proc:
f(0):= 0: f(1):= 1: f(2):= 12: f(3):= 13:
seq(f(i),i=0..100); # Robert Israel, Oct 21 2016

a(n) = my(ret=0,p=0); while(n, ret+=[0,1,12,13][n%4+1]<Kevin Ryde, Sep 07 2019

Perl
```
See Links section.
```

from gmpy2 import c_divmod
u = ('0000','1000','0011','1011')
def A066321(n):
    if n == 0:
        return 0
    else:
        s, q = '', n
        while q:
            q, r = c_divmod(q, -4)
            s += u[r]
        return int(s[::-1],2) # Chai Wah Wu, Apr 09 2016

In "rebase notation" a(n) = (i-1)[n]2.

G.f. g(z) satisfies g(z) = z*(1+12*z+13*z^2)/(1-z^4) + 16*z^4*(13+12*z^4+z^8)/((1-z)*(1+z^4)*(1+z^8)) + 256*(1-z^16)*g(z^16)/(z^12-z^13). - Robert Israel, Oct 21 2016

A320283 Lexicographical ordering of pure imaginary integers in the base (-1+i) numeral system.

Original entry on oeis.org

0, 1, -2, -1, -4, -3, -6, -5, 8, 9, 6, 7, 4, 5, 2, 3, 16, 17, 14, 15, 12, 13, 10, 11, 24, 25, 22, 23, 20, 21, 18, 19, -32, -31, -34, -33, -36, -35, -38, -37, -24, -23, -26, -25, -28, -27, -30, -29, -16, -15, -18, -17, -20, -19, -22, -21, -8, -7, -10, -9, -12, -11, -14, -13, -64, -63, -66, -65, -68, -67, -70, -69

Offset: 0

Andreas K. Badea, Oct 09 2018

For ordering of pure real integers in same system see A073791.

All integers appear in this sequence.

Andrey Zabolotskiy, Table of n, a(n) for n = 0..8191 (terms up to 255 from Andreas K. Badea)
Solomon I. Khmelnik, Specialized Digital Computer for Operations with Complex Numbers, Questions of Radio Electronics, 12 (1964), 60-82 [in Russian].
W. J. Penney, A "binary" system for complex numbers, NSA Technical Journal, Vol. X, No. 2 (1965), 13-15.
W. J. Penney, A "binary" system for complex numbers, JACM 12 (1965), 247-248.

Cf. A066321, A256441, A073791.

From Andrey Zabolotskiy, Jan 31 2019: (Start)
a(n) = A073791(2*n)/2.
a(n) = -a(4*n)/4.
a(n) = -4*a(floor(n/4)) + a(n mod 4). (End)

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.

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A066321 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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A320283 Lexicographical ordering of pure imaginary integers in the base (-1+i) numeral system.

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

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