A177505 Base 2i representation of n reinterpreted in base 4.

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

Offset: 0

Author

Alonso del Arte, Feb 03 2012

Keywords

• nonn
• easy
• base

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 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).

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.

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

Examples

			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.
		

References

• Donald Knuth, The Art of Computer Programming. Volume 2, 2nd Edition. Reading, Massachussetts: Addison-Wesley (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. 245-247.
• OEIS Wiki, Quater-imaginary base
• Wikipedia, Quater-imaginary base

Crossrefs

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

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

Programs

Mathematica

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

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

(End)