A239200 - OEIS

%I #12 Mar 12 2014 10:49:01

%S 1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,1,0,0,1,0,1,1,1,

%T 0,0,0,0,0,0,0,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0,1,0,1,1,1,0,0,0,0,1,1,

%U 0,0,0,1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,0,0,1,1,1

%N Expansion of Pi in the irrational base b=log(7).

%C 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.

%C 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).

%H George Bergman, <a href="http://www.jstor.org/stable/3029218">A number system with an irrational base</a>, Math. Mag. 31 (1957), pp. 98-110 (available on JSTOR.org).

%e 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).

%o (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}

%o base(Pi) \\ returns this sequence as a vector (whose components are indexed by 1,2,3... instead of -1,0,1,...).

%Y Cf. A238897 (Pi in base sqrt(2)), A239199 (Pi in base sqrt(3)), A050948 (Pi in base e), A050949 (e in base Pi), A102243 (Pi in the golden base).

%K nonn,base

%O -1

%A _M. F. Hasler_, Mar 12 2014