A ternary notation

[Franklin T. Adams-Watters]

"Eric Angelini" <keynews.tv at skynet.be> wrote:
> Hello SeqFan and Math-Fun,
> Consider this (hope this is not old-hat):
>
>   3^0  3^1  3^2  3^3  3^4  3^5  3^6  3^7 ...
>=   1    3    9    27   81  243  729  2187 ...
>
>This seq. is known to be very efficient if you
>want to weigh integer weights with a two-tray
>balance, leaving no "holes" behind; use weights
>of 1,3,9,27... units to measure all "natural"
>quantities, from 1 to infinity:
>
>I had the idea, yesterday night, to represent
>all natural numbers in the same way -- all I
>had to do was to use three symbols :
>  0 for a power of 3 I don't need
>  1 for a power of 3 I need
>  2 as a grahic symbol meaning "minus all the
>    rest"

This is an old idea, but there is a conceptually much simpler approach to the third digit: it is the digit -1, representing putting just this weight in the other pan.  The result is called "balanced ternary" - I first encountered it in Knuth's The Art of Computer Programming (I forget which volume).  If you use 2 to represent the -1 digit, you will get:
0, 1, 12, 10, 11, 122, 120, 121, 102, 100, 101, ...

This sequence is also not in the OEIS.
-- 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645


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