A239199 Expansion of Pi in the irrational base b=sqrt(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

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. 98-110.

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>0||break); 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: A373139 A122276 A352679 * A265718 A267463 A264442

Adjacent sequences: A239196 A239197 A239198 * A239200 A239201 A239202

Keyword

nonn,base

Author

M. F. Hasler, Mar 12 2014

Status

approved