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A059095 List consisting of the representation of n in base 3 using digits -1, 0, 1. 27
1, 1, -1, 1, 0, 1, 1, 1, -1, -1, 1, -1, 0, 1, -1, 1, 1, 0, -1, 1, 0, 0, 1, 0, 1, 1, 1, -1, 1, 1, 0, 1, 1, 1, 1, -1, -1, -1, 1, -1, -1, 0, 1, -1, -1, 1, 1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, 1, 1, -1, 1, -1, 1, -1, 1, 0, 1, -1, 1, 1, 1, 0, -1, -1, 1, 0, -1, 0, 1, 0, -1, 1, 1, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, -1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, -1, -1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1

COMMENTS

Every natural number n has a unique representation as n = Sum_{i=1..k} e(i)*(3^i) for some k where e(i) is one of -1,0,1. Example: 25 = 27-3+1 = 1*3^3+0*3^2+(-1)*3^1+1*3^0 so its representation is 1,0,-1,1. So by writing n in this base 3 representation and juxtaposing we get the sequence: (1), (1,-1), (1,0), (1,1), (1,-1,-1), ...

REFERENCES

D. E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, Vol 2, pp 173-175.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..4560 (rows 1 <= n <= 729 = 3^6, flattened)

Wikipedia, Balanced Ternary

FORMULA

n = Sum_{0 <= k < A134021(n)} a(A134421(n)-2-k)*3^k, for n>0. - Reinhard Zumkeller, Oct 25 2007

EXAMPLE

From Michael De Vlieger, Jun 27 2020: (Begin)

First 27 rows, with terms aligned with powers of 3:

3^3 3^2 3^1 3^0

--------------------

1: 1;

2: 1, -1;

3: 1, 0;

4: 1, 1;

5: 1, -1, -1;

6: 1, -1, 0;

7: 1, -1, 1;

8: 1, 0, -1;

9: 1, 0, 0;

10: 1, 0, 1;

11: 1, 1, -1;

12: 1, 1, 0;

13: 1, 1, 1;

14: 1, -1, -1, -1;

15: 1, -1, -1, 0;

16: 1, -1, -1, 1;

17: 1, -1, 0, -1;

18: 1, -1, 0, 0;

19: 1, -1, 0, 1;

20: 1, -1, 1, -1;

21: 1, -1, 1, 0;

22: 1, -1, 1, 1;

23: 1, 0, -1, -1;

24: 1, 0, -1, 0;

25: 1, 0, -1, 1;

26: 1, 0, 0, -1;

27: 1, 0, 0, 0;

... (End)

MATHEMATICA

Array[If[First@ # == 0, Rest@ #, #] &[Prepend[IntegerDigits[#, 3], 0] //. {a___, b_, 2, c___} :> {a, b + 1, -1, c}] &, 32] // Flatten (* Michael De Vlieger, Jun 27 2020 *)

PROG

(Python)

def b3(n):

if n == 0: return []

carry, trailing = [(0, 0), (0, 1), (1, -1)][n % 3]

return b3(n//3 + carry) + [trailing]

t = []

for n in range(50):

t += b3(n)

print(t)

# Andrey Zabolotskiy, Nov 10 2019

(PARI) row(n) = apply(d->d-1, digits(n + 3^(logint(n<<1, 3)+1)>>1, 3)); \\ Kevin Ryde, Mar 04 2022

CROSSREFS

Cf. A117966, A134021 (row lengths, starting from row 1), A102283 (last each row), A065363 (row sums).

Cf. A003137 (ternary).

Sequence in context: A099990 A089939 A330550 * A187944 A105597 A188470

Adjacent sequences: A059092 A059093 A059094 * A059096 A059097 A059098

KEYWORD

tabf,sign,easy

AUTHOR

Avi Peretz (njk(AT)netvision.net.il), Feb 13 2001

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2001

Offset corrected by Andrey Zabolotskiy, Nov 10 2019

STATUS

approved

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Last modified February 9 02:59 EST 2023. Contains 360153 sequences. (Running on oeis4.)