OFFSET
-1
COMMENTS
The negative offset is chosen as to have Pi = sum(a(i)*b^-i, i=offset...+oo), with the base b=log(7), cf. Example.
Log(7) is the largest base of the form log(n) < 2, such that the expansion has only digits 1 and 0 (and can therefore also be recorded in a condensed way by just listing the positions of nonzero digits, cf. example). Sqrt(3) has this maximal property for bases of the form sqrt(n).
LINKS
George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), pp. 98-110 (available on JSTOR.org).
EXAMPLE
Pi = log(7)^1 + log(7)^0 + log(7)^-3 + log(7)^-5 + ... = [1,1;0,0,1,0,1,1,...]_{log(7)}, which could also be encoded as (1,0,-3,-5,...) or (-1,0,3,5,...) (sequence of which the present one is the characteristic function).
PROG
(PARI) base(x, b=log(7), L=99, a=[])={ forstep(k=log(x)\log(b), -L, -1, a=concat(a, d=x\b^k); (x-=d*b^k)||k>0||break); a}
base(Pi) \\ returns this sequence as a vector (whose components are indexed by 1, 2, 3... instead of -1, 0, 1, ...).
CROSSREFS
KEYWORD
nonn,base
AUTHOR
M. F. Hasler, Mar 12 2014
STATUS
approved