OFFSET
0,3
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
PROG
(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
(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)
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
N. J. A. Sloane, Apr 08 2016
STATUS
approved