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A212494 Base 2i representation of nonnegative integers.

%I A212494 #31 Apr 06 2020 20:26:21
%S A212494 0,1,2,3,10300,10301,10302,10303,10200,10201,10202,10203,10100,10101,
%T A212494 10102,10103,10000,10001,10002,10003,20300,20301,20302,20303,20200,
%U A212494 20201,20202,20203,20100,20101,20102,20103
%N A212494 Base 2i representation of nonnegative integers.
%C A212494 The use of negabinary dispenses with the need for sign bits and for keeping track of signed and unsigned data types. Similarly, the use of base 2i, or quater-imaginary, dispenses with the need to represent the real and imaginary parts of a complex number separately. (The term "quater-imaginary" appears in Knuth's landmark book on computer programming).
%C A212494 Quater-imaginary, based on the powers of 2i (twice the imaginary unit), uses the digits 0, 1, 2, 3. For purely real positive integers, the quater-imaginary representation is the same as negaquartal (base -4) except that 0's are "riffled" in, corresponding to the odd-indexed powers of 2i which are purely imaginary numbers. Therefore, to obtain the base 2i representations of positive real numbers, the algorithm for base -4 representations can be employed with only a small adjustment.
%C A212494 To obtain the base 2i representation of a complex number a+bi, do as above for the real part, then again for the real part of 2i*(a+bi) = -2b+2ai, giving the digits corresponding to the odd-indexed powers of 2i.
%C A212494 Omitting digits for odd powers of 2i (all 0's for the imaginary parts) (e.g. 20300 --> 230) gives A007608 (nonnegative integers in base -4).
%D A212494 Donald Knuth, The Art of Computer Programming. Volume 2, 2nd Edition. Reading, Massachussetts: Addison-Wesley (1981): 189
%H A212494 Joerg Arndt, <a href="/A212494/b212494.txt">Table of n, a(n) for n = 0..1000</a>
%H A212494 Joerg Arndt, <a href="http://www.jjj.de/fxt/demo/bits/#radix-2i">Radix 2i</a>
%H A212494 Donald Knuth, <a href="http://dl.acm.org/citation.cfm?id=367233">An imaginary number system</a>, Communications of the ACM 3 (4), April 1960, pp. 245-247.
%H A212494 OEIS Wiki, <a href="/wiki/Quater-imaginary_base">Quater-imaginary base</a>
%H A212494 Wikipedia, <a href="http://en.wikipedia.org/wiki/Quater-imaginary_base">Quater-imaginary base</a>
%e A212494 a(5) = 10301 because 5 = 1*(2i)^4+3*(2i)^2+1*(2i)^0 = 1*16+3*(-4)+1*1
%Y A212494 Cf. A212542 (Base 2i representation of negative integers).
%Y A212494 Cf. A177505.
%Y A212494 Cf. A007608 (Nonnegative integers in base -4).
%K A212494 nonn,base
%O A212494 0,3
%A A212494 _Daniel Forgues_, May 18 2012