A177505 - OEIS

%I #47 Apr 07 2020 21:35:07

%S 0,1,2,3,304,305,306,307,288,289,290,291,272,273,274,275,256,257,258,

%T 259,560,561,562,563,544,545,546,547,528,529,530,531,512,513,514,515,

%U 816,817,818,819,800,801,802,803,784,785

%N Base 2i representation of n reinterpreted in base 4.

%C 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 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 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. - _Daniel Forgues_, May 18 2012

%D Donald Knuth, The Art of Computer Programming. Volume 2, 2nd Edition. Reading, Massachussetts: Addison-Wesley (1981): 189

%H Vincenzo Librandi, <a href="/A177505/b177505.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 OEIS Wiki, <a href="/wiki/Quater-imaginary_base">Quater-imaginary base</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Quater-imaginary_base">Quater-imaginary base</a>

%F Conjectures from _Colin Barker_, Jul 16 2019: (Start)

%F G.f.: x*(1 + x + x^2 + 301*x^3 + x^4 + x^5 + x^6 - 19*x^7 + x^8 + x^9 + x^10 - 19*x^11 + x^12 + x^13 + x^14 - 19*x^15) / ((1 - x)^2*(1 + x)*(1 + x^2)*(1 + x^4)*(1 + x^8)).

%F a(n) = a(n-1) + a(n-16) - a(n-17) for n>16.

%F (End)

%e a(5) = 305 because 5 in base 2i is 10301 ( = (2i)^4 + 3 * (2i)^2 + (2i)^0), and (-4)^4 + 3 * (-4)^2 + (-4)^0 = 256 + 3 * 16 + 1 = 305.

%t (* First run the program from A039724 to define ToNegaBases *) Table[FromDigits[Riffle[IntegerDigits[ToNegaBases[n, 4]], 0], 4], {n, 0, 63}]

%Y Cf. A005351 (base -2 representation of n reinterpreted as binary).

%Y Cf. A212494 (base 2i representation of n).

%K nonn,easy,base

%O 0,3

%A _Alonso del Arte_, Feb 03 2012