A134024 -id:A134024 - cp's OEIS frontend

A065363 Sum of balanced ternary digits in n. Replace 3^k with 1 in balanced ternary expansion of n.

0, 1, 0, 1, 2, -1, 0, 1, 0, 1, 2, 1, 2, 3, -2, -1, 0, -1, 0, 1, 0, 1, 2, -1, 0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 2, 1, 2, 3, 2, 3, 4, -3, -2, -1, -2, -1, 0, -1, 0, 1, -2, -1, 0, -1, 0, 1, 0, 1, 2, -1, 0, 1, 0, 1, 2, 1, 2, 3, -2, -1, 0, -1, 0, 1, 0, 1, 2, -1, 0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 2, 1, 2, 3, 2, 3, 4, -1, 0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 2, 1, 2

Offset: 0

Marc LeBrun, Oct 31 2001

Notation: (3)(1).
Extension to negative n: a(-n) = -a(n). - Franklin T. Adams-Watters, May 13 2009
Row sums of A059095. - Rémy Sigrist, Oct 05 2019

			5 = + 1(9) - 1(3) - 1(1) -> +1 - 1 - 1 = -1 = a(5).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6.
Daniel Forgues, Table of n, a(n) for n = 0..100000

Cf. A059095, A065364, A053735. See A134452 for iterations.

Cf. A134022, A134024.

a:= proc(n) `if`(n=0, 0, (d-> `if`(d=2,
      a(q+1)-1, d+a(q)))(irem(n, 3, 'q')))
    end:
seq(a(n), n=0..120);  # Alois P. Heinz, Jan 09 2020

balTernDigits[0] := {0}; balTernDigits[n_/;n > 0] := Module[{unParsed = n, currRem, currExp = 1, digitList = {}, nextDigit}, While[unParsed > 0, If[unParsed == 3^(currExp - 1), digitList = Append[digitList, 1]; unParsed = 0, currRem = Mod[unParsed, 3^currExp]/3^(currExp - 1); nextDigit = Switch[currRem, 0, 0, 2, -1, 1, 1]; digitList = Append[digitList, nextDigit]; unParsed = unParsed - nextDigit * 3^(currExp - 1)]; currExp++]; digitList = Reverse[digitList]; Return[digitList]]; balTernDigits[n_/;n < 0] := (-1)balTernDigits[Abs[n]]; Table[Plus@@balTernDigits[n], {n, 0, 108}] (* Alonso del Arte, Feb 25 2011 *)
terVal[lst_List] := Reverse[lst].(3^Range[0, Length[lst] - 1]); maxDig = 4; t = Table[0, {3 * 3^maxDig/2}]; t[[1]] = 1; Do[d = IntegerDigits[Range[0, 3^dig - 1], 3, dig]/.{2 -> -1}; d = Prepend[#, 1]&/@d; t[[terVal/@d]] = Total/@d, {dig, maxDig}]; Prepend[t, 0] (* T. D. Noe, Feb 24 2011 *)
Array[Total[Prepend[IntegerDigits[#, 3], 0] //. {a___, b_, 2, c___} :> {a, b + 1, -1, c}] &, 109, 0] (* Michael De Vlieger, Jun 27 2020 *)

bt(n)=my(d=digits(n,3),c=1); while(c, if(d[1]==2, d=concat(0,d)); c=0; for(i=2,#d, if(d[i]==2,d[i]=-1;d[i-1]+=1;c=1))); d
a(n)=vecsum(bt(n)) \\ Charles R Greathouse IV, May 07 2020

def a(n):
    s=0
    x=0
    while n>0:
        x=n%3
        n=n//3
        if x==2:
            x=-1
            n+=1
        s+=x
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 06 2017

G.f.: (1/(1-x))*Sum_{k>=0} (x^3^k - x^(2*3^k))/(x^((3^k-1)/2)*(1 + x^3^k + x^(2*3^k))). - Franklin T. Adams-Watters, May 13 2009
a(n) = A134024(n) - A134022(n). - Reinhard Zumkeller, Dec 16 2010
a(3*n - 1) = a(n) - 1, a(3*n) = a(n), a(3*n + 1) = a(n) + 1. - Thomas König, Jun 24 2020
a(n) = A053735(2n) - A053735(n). This can be shown with constructing balanced ternary representation of n: Add n and n with carry, and then subtract n from the sum without borrow. - Yifan Xie, Dec 24 2024

A134021 Length of n in balanced ternary representation.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Reinhard Zumkeller, Oct 19 2007

Shifted variant of A064099.

			100 = 1*3^4+1*3^3-1*3^2+0*3^1+1*3^0: a(100) = |++-0+| = 5.
200 = 1*3^5-1*3^4+1*3^3+1*3^2+1*3^1-1*3^0: a(200) = |+-+++-| = 6.
300 = 1*3^5+1*3^4-1*3^3+0*3^2+1*3^1+0*3^0: a(300) = |++-0+0| = 6.

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Wikipedia, Balanced Ternary.

Cf. A005812, A059095, A081604, A134022, A134023, A134024, A134025, A134026, A134027, A134028, A134421.

a[n_] := Ceiling[Log[3, 2*n+1]]; a[0] = 1; Array[a, 100, 0] (* Amiram Eldar, Apr 03 2025 *)

def a(n):
    if n==0: return 1
    s=0
    x=0
    while n>0:
        x=n%3
        n=n//3
        if x==2:
            x=-1
            n+=1
        s+=1
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

For n > 0: a(n) = ceiling(log(2*n+1)/log(3)).
a(n) = A134022(n) + A134023(n) + A134024(n).
0 <= a(n) - A081604(n) <= 1.
a(A134025(n)) = A081604(A134025(n)); a(A134026(n)) = A081604(A134026(n))+1.
a(A134027(n)) = a(n); a(abs(A134028(n))) <= a(n).
a(n) = A064099(n-1) for n>1.
n = Sum_{k=0..a(n)-1} (A059095(A134421(n)-2-k)*3^k), for n > 0. - Reinhard Zumkeller, Oct 25 2007
a(n) = A005812(n) + A134023(n).

A005812 Weight of balanced ternary representation of n.

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 3, 4, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 3

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

Weight of n means count of nonzero digits of n. - Daniel Forgues, Mar 24 2010

a(n) = A134022(n) + A134024(n) = A134021(n) - A134023(n).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n = 0..100000
P. Flajolet and Lyle Ramshaw, A note on Gray code and odd-even merge, SIAM J. Comput. 9 (1980), 142-158.
Michael Gilleland, Some Self-Similar Integer Sequences

(defun btw (n) (if (= n 0) 0 (multiple-value-bind (q r) (round n 3) (+ (abs r) (btw q)))))

a[n_] := With[{q=Round[n/3]}, Abs[n-3q]+a[q]]; a[0]=0; Table[a[n], {n, 0, 105}](* Jean-François Alcover, Nov 25 2011, after Pari *)

a(n)=local(q); if(n<=0,0,q=round(n/3); abs(n-3*q)+a(q))

def a(n):
    s=0
    x=0
    while n>0:
        x=n%3
        n//=3
        if x==2:
            x=-1
            n+=1
        if x!=0: s+=1
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

a(3n)=a(n), a(3n+1)=a(n)+1, a(9n+2)=a(n)+2, a(9n+5)=a(3n+2)+1, a(9n+8)=a(3n+2).

a(n) = Sum_{k>0} floor(|2*sin(n*Pi/3^k)|). - Toshitaka Suzuki, Sep 10 2006

Additional terms from Allan C. Wechsler

A134023 Number of zeros in balanced ternary representation of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1

Offset: 0

Reinhard Zumkeller, Oct 19 2007

			100=1*3^4+1*3^3-1*3^2+0*3^1+1*3^0=='++-0+': a(100)=1;
200=1*3^5-1*3^4+1*3^3+1*3^2+1*3^1-1*3^0=='+-+++-': a(200)=0;
300=1*3^5+1*3^4-1*3^3+0*3^2+1*3^1+0*3^0=='++-0+0': a(300)=2.

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Wikipedia, Balanced Ternary

Cf. A005812, A059095, A134021, A134022, A134024.

Array[Count[If[First@ # == 0, Rest@ #, #], 0] &[Prepend[IntegerDigits[#, 3], 0] //. {a___, b_, 2, c___} :> {a, b + 1, -1, c}] &, 105, 0] (* Michael De Vlieger, Jun 27 2020 *)

def a(n):
    if n==0: return 1
    s=0
    x=0
    while n>0:
        x=n%3
        n=n//3
        if x==2:
            x=-1
            n+=1
        if x==0: s+=1
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

a(n) = A134021(n) - A134022(n) - A134024(n).

a(n) = A134021(n) - A005812(n).

A134022 Number of negative trits in balanced ternary representation of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 4, 3, 3, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 1, 1, 2, 1, 1, 3, 2, 2, 2, 1, 1, 2, 1, 1, 3, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 2, 1, 1, 2, 1, 1, 2

Offset: 0

Reinhard Zumkeller, Oct 19 2007

			100 = 1*3^4+1*3^3-1*3^2+0*3^1+1*3^0 == '++-0+': a(100) = 1;
200 = 1*3^5-1*3^4+1*3^3+1*3^2+1*3^1-1*3^0 == '+-+++-': a(200) = 2;
300 = 1*3^5+1*3^4-1*3^3+0*3^2+1*3^1+0*3^0 == '++-0+0': a(300) = 1.

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Wikipedia, Balanced Ternary

Cf. A059095, A134023, A134024.

Array[Count[#, -1] &[Prepend[IntegerDigits[#, 3], 0] //. {a___, b_, 2, c___} :> {a, b + 1, -1, c}] &, 105, 0] (* Michael De Vlieger, Jun 27 2020 *)

def a(n):
    s=0
    x=0
    while n>0:
        x=n%3
        n=n//3
        if x==2:
            x=-1
            n+=1
        if x==-1: s+=1
    return s
print([a(n) for n in range(151)]) # Indranil Ghosh, Jun 07 2017

a(n) = A134021(n) - A134023(n) - A134024(n).

a(n) = A005812(n) - A134024(n) = A134024(n) - A065363(n).

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A065363 Sum of balanced ternary digits in n. Replace 3^k with 1 in balanced ternary expansion of n.

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A134021 Length of n in balanced ternary representation.

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A005812 Weight of balanced ternary representation of n.

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A134023 Number of zeros in balanced ternary representation of n.

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A134022 Number of negative trits in balanced ternary representation of n.

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