# [seqfan] Complex base reinterpretation sequence

Pierre Abbat phma at bezitopo.org
Sun May 15 09:46:12 CEST 2022

```I just made the following sequence:

> take 100 (map (gosperSerial . integerToGosperInteger) [0..])
[0,1,9,10,179,180,188,182,183,1269,1270,1243,1244,1252,1246,1247,1059,1060,
1033,1034,1042,1036,1037,1094,1095,1215,1216,9505,9499,9500,9459,9460,9482,
9483,9295,9289,9290,9298,9299,9272,9273,9330,9324,9325,9333,9334,8131,8132,
8091,8085,8086,8094,8095,7921,7922,7930,7924,7925,7982,7983,7956,7957,7965,
7959,7960,47560,47561,47534,47535,47543,47537,47538,47595,47596,47373,47374,
47431,47425,47426,47385,47386,47408,47409,46192,46186,46187,46195,46196,
46169,46170,46227,46221,46222,46230,46231,46057,46058,46017,46011,46012]

The steps to compute the nth number in the sequence are:
1. Express n as an Eisenstein integer in base 2.5-√-0.75 (called the Gosper
base) using cyclotomic digits, where 0 and 1 represent themselves and the
units are 1,3,2,6,4,5 counterclockwise.
2. Reinterpret the sequence of digits as base 7.
3. Output the resulting nonnegative integer in decimal.

The growth is on the order of quadratic, but with big jumps both up and down.
For all n, a[n] ≡ n mod 7.

The program that generates this sequence is at https://github.com/phma/
gosperbase . Like Quadlods, which generates A322289 but is a library and
application for generating two kinds of low-discrepancy sequences, Gosperbase,
which I'm still developing, will be a library for computing both integers and
floats in Gosper base.

I don't expect any applied use of Gosperbase. It's just a demonstration that
arithmetic, including long division, can be done in a complex base. (I've done
the division code in C++, but then got stuck on the parser, so I'm rewriting
the whole thing in Haskell.) Quadlods is intended for applied use; I currently
have a job prospect which may use Quadlods.

Pierre
--
gau do li'i co'e kei do

```