A331990 Balanced ternary expansion of e.

1, 0, -1, 0, 1, 1, 1, -1, -1, 0, -1, 0, -1, 1, 1, 1, -1, 0, 1, 1, 1, -1, 0, 0, 0, -1, 1, 1, -1, 0, 1, 1, 1, 0, -1, 0, 0, 0, -1, 1, -1, -1, 1, 0, 0, 0, 0, -1, 1, 1, 0, -1, 0, 0, -1, -1, 1, -1, 1, -1, -1, 0, -1, -1, 0, 0, -1, 1, 1, 1, 0, 0, 1, -1, -1, 1, 0, 0, 0

Offset

2

Comments

For the first n digits of the sequence, take the smallest substring with n or more digits of the standard ternary expansion of e (A004594) such that it does not end with 1 and prepend a 0. For n = 15, this would be:

0, 2, 2, 0, 1, 1, 0, 1, 1, 2, 1, 2, 2, 1, 1, 0.

Wherever there is a 2, change it to -1 and add 1 to the previous term:

1, 0, -1, 0, 1, 1, 0, 1, 2, -1, 2, 0, -1, 1, 1, 0.

Repeat until no 2's remain:

1, 0, -1, 0, 1, 1, 0, 2, -1, 0, -1, 0, -1, 1, 1, 0.

1, 0, -1, 0, 1, 1, 1, -1, -1, 0, -1, 0, -1, 1, 1, 0.

Remove all but the first n digits:

1, 0, -1, 0, 1, 1, 1, -1, -1, 0, -1, 0, -1, 1, 1.

Links

Iain Fox, Table of n, a(n) for n = 2..20000

Wikipedia, Balanced ternary

Example

e = 2.7182818284... = 1 * 3^1 + 0 * 3^0 - 1 * 3^(-1) + 0 * 3^(-2) + 1 * 3^(-3) + ... = 10.T0111TT0T0..._bal3

Prog

(PARI) first(n) = {default(realprecision, 10000); for(x=-1, +oo, v=concat([0], digits(floor(exp(1)*3^(n+x)), 3)); if(v[#v]!=1, break())); while(vecmax(v)==2, for(x=1, #v, if(v[x]==2, v[x]=-1; v[x-1]++))); vecextract(v, 2^n-1)} \\ (adjust realprecision as needed)

Crossrefs

Pi in balanced ternary: A331313.

Expansion of e in base b: A004593 (b = 2), A004594 (b = 3), A004595 (b = 4), A004596 (b = 5), A004597 (b = 6), A004598 (b = 7), A004599 (b = 8), A004600 (b = 9), A001113 (b = 10), A170873 (b = 16).

Sequence in context: A244527 A243566 A266112 * A301849 A074332 A152065

Adjacent sequences: A331987 A331988 A331989 * A331991 A331992 A331993

Keyword

base,easy,sign

Author

Iain Fox, Feb 03 2020

Status

approved