

A177505


Base 2i representation of n reinterpreted in base 4.


2



0, 1, 2, 3, 304, 305, 306, 307, 288, 289, 290, 291, 272, 273, 274, 275, 256, 257, 258, 259, 560, 561, 562, 563, 544, 545, 546, 547, 528, 529, 530, 531, 512, 513, 514, 515, 816, 817, 818, 819, 800, 801, 802, 803, 784, 785
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

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 quaterimaginary, dispenses with the need to represent the real and imaginary parts of a complex number separately. (The term "quaterimaginary" appears in Knuth's landmark book on computer programming).
Quaterimaginary, based on the powers of 2i (twice the imaginary unit), uses the digits 0, 1, 2, 3. For purely real positive integers, the quaterimaginary representation is the same as negaquartal (base 4) except that 0s are "riffled" in, corresponding to the oddindexed 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.
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 oddindexed powers of 2i.  Daniel Forgues, May 18 2012


REFERENCES

Donald Knuth, The Art of Computer Programming. Volume 2, 2nd Edition. Reading, Massachussetts: AddisonWesley (1981): 189


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Donald Knuth, An imaginary number system, Communications of the ACM 3 (4), April 1960, pp. 245247.
OEIS Wiki, Quaterimaginary base
Wikipedia, Quaterimaginary base


FORMULA

Conjectures from Colin Barker, Jul 16 2019: (Start)
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)).
a(n) = a(n1) + a(n16)  a(n17) for n>16.
(End)


EXAMPLE

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.


MATHEMATICA

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


CROSSREFS

Cf. A005351 (base 2 representation of n reinterpreted as binary).
Cf. A212494 (base 2i representation of n).
Sequence in context: A087313 A004876 A307212 * A068104 A065586 A110931
Adjacent sequences: A177502 A177503 A177504 * A177506 A177507 A177508


KEYWORD

nonn,easy,base


AUTHOR

Alonso del Arte, Feb 03 2012


STATUS

approved



