A271472 Binary representation of n in base i-1.

0, 1, 1100, 1101, 111010000, 111010001, 111011100, 111011101, 111000000, 111000001, 111001100, 111001101, 100010000, 100010001, 100011100, 100011101, 100000000, 100000001, 100001100, 100001101, 110011010000, 110011010001, 110011011100, 110011011101, 110011000000, 110011000001

Offset: 0

Author

N. J. A. Sloane, Apr 08 2016

Keywords

• nonn
• base
• easy

Comments

This is A066321 converted from base 10 to base 2.

Every Gaussian integer r+s*i (r, s ordinary integers) has a unique representation as a sum of powers of t = i-1. For example 3 = 1+b^2+b^3, that is, "1101" in binary, which explains a(3) = 1101. See A066321 for further information.

From Jianing Song, Jan 22 2023: (Start)

Also binary representation of n in base -1-i.

Write out n in base -4 (A007608), then change each digit 0, 1, 2, 3 to 0000, 0001, 1100, 1101 respectively. (End)

References

• 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.)
• W. J. Penney, A "binary" system for complex numbers, JACM 12 (1965), 247-248.

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000
• N. J. A. Sloane, Table of n, (I-1)^n for n=0..100
• Wikipedia, Complex-base system

Crossrefs

Cf. A066321.

Programs

PARI

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)

Python

from gmpy2 import c_divmod
u = ('0000','1000','0011','1011')
def A271472(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]) # Chai Wah Wu, Apr 09 2016