A239200 Expansion of Pi in the irrational base b=log(7).

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, 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, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1

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

Table of n, a(n) for n=-1..90.

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

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).

Sequence in context: A328102 A177444 A325896 * A157686 A181115 A284527

Adjacent sequences: A239197 A239198 A239199 * A239201 A239202 A239203

Keyword

nonn,base

Author

M. F. Hasler, Mar 12 2014

Status

approved