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%I #17 Mar 18 2014 14:20:42
%S 1,0,0,0,0,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,1,1,
%T 0,0,0,1,1,0,0,0,0,0,1,0,1,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,
%U 1,0,1,0,1,0,0,1,1,0,0,0,0,1,0,1,0,1,0,0,0,0,0,1,0,1,0,1,0,0,0,1,0,0
%N Expansion of Pi in the irrational base b=sqrt(3).
%C The negative offset is chosen as to have Pi = sum(a(i)*b^-i, i=offset...+oo), with the base b=sqrt(3), cf. Example.
%C Sqrt(3) is the largest base of the form sqrt(n) < 2, such that the expansion of any number in this base has only digits 1 and 0 (which allows a condensed version of the expansion which lists only the positions of the nonzero digits, here: -2, 4, 7, 9, 12, 14, 17, 18, 24, 26, ...). Log(7) has this maximal property for bases of the form log(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.
%e Pi = sqrt(3)^2 + sqrt(3)^-4 + sqrt(3)^-7 + ... = [1,0,0;0,0,0,1,0,0,1,...]_{sqrt(3)}.
%t RealDigits[Pi, Sqrt[3], 105][[1]] (* _T. D. Noe_, Mar 12 2014 *)
%o (PARI) base(x,b=sqrt(3),L=99/*max.# digits for fract.part*/,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 A239199 = base(Pi) \\ defines A239199 as a vector; indices are here 1,2,3... instead of -2,-1,0,....
%Y Cf. A238897 (Pi in base sqrt(2)), A050948 (Pi in base e), A050949 (e in base Pi), A102243 (Pi in the golden base).
%K nonn,base
%O -2
%A _M. F. Hasler_, Mar 12 2014
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