

A239199


Expansion of Pi in the irrational base b=sqrt(3).


3



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, 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, 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
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OFFSET

2


COMMENTS

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.
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).


LINKS

Table of n, a(n) for n=2..99.
George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), pp. 98110.


EXAMPLE

Pi = sqrt(3)^2 + sqrt(3)^4 + sqrt(3)^7 + ... = [1,0,0;0,0,0,1,0,0,1,...]_{sqrt(3)}.


MATHEMATICA

RealDigits[Pi, Sqrt[3], 105][[1]] (* T. D. Noe, Mar 12 2014 *)


PROG

(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>0break); a}
A239199 = base(Pi) \\ defines A239199 as a vector; indices are here 1, 2, 3... instead of 2, 1, 0, ....


CROSSREFS

Cf. A238897 (Pi in base sqrt(2)), A050948 (Pi in base e), A050949 (e in base Pi), A102243 (Pi in the golden base).
Sequence in context: A089010 A162289 A122276 * A265718 A267463 A264442
Adjacent sequences: A239196 A239197 A239198 * A239200 A239201 A239202


KEYWORD

nonn,base


AUTHOR

M. F. Hasler, Mar 12 2014


STATUS

approved



