A134028 Reversal of n in balanced ternary representation.

0, 1, -2, 1, 4, -11, -2, 7, -8, 1, 10, -5, 4, 13, -38, -11, 16, -29, -2, 25, -20, 7, 34, -35, -8, 19, -26, 1, 28, -17, 10, 37, -32, -5, 22, -23, 4, 31, -14, 13, 40, -119, -38, 43, -92, -11, 70, -65, 16, 97, -110, -29, 52, -83, -2, 79, -56, 25, 106, -101, -20, 61, -74, 7, 88, -47, 34, 115, -116, -35, 46, -89, -8, 73, -62, 19, 100

Offset: 0

Author

Reinhard Zumkeller, Oct 19 2007

Keywords

• sign
• look
• base

Comments

As the graph demonstrates, the sequence makes large negative steps at terms (3^i+1)/2. These steps divide the graph into conspicuous blocks. - N. J. A. Sloane, Jul 03 2016

Examples

			20 = 1*3^3 - 1*3^2 + 1*3^1 - 1*3^0 == '+-+-'
=> a(20) = -1*3^3 + 1*3^2 - 1*3^1 + 1*3^0 = -20;
21 = 1*3^3 - 1*3^2 + 1*3^1 + 0*3^0 == '+-+0'
=> a(21) = 0*3^3 + 1*3^2 - 1*3^1 + 1*3^0 = 7;
22 = 1*3^3 - 1*3^2 + 1*3^1 + 1*3^0 == '+-++'
=> a(22) = 1*3^3 + 1*3^2 - 1*3^1 + 1*3^0 = 34;
23 = 1*3^3 + 0*3^2 - 1*3^1 - 1*3^0 == '+0--'
=> a(23) = -1*3^3 - 1*3^2 + 0*3^1 + 1*3^0 = -35;
24 = 1*3^3 + 0*3^2 - 1*3^1 + 0*3^0 == '+0-0'
=> a(24) = 0*3^3 - 1*3^2 + 0*3^1 + 1*3^0 = -8;
25 = 1*3^3 + 0*3^2 - 1*3^1 + 1*3^0 == '+0-+'
=> a(25) = 1*3^3 - 1*3^2 + 0*3^1 + 1*3^0 = 19.
		

References

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Reversal
• Wikipedia, Balanced Ternary

Crossrefs

Cf. A117966 (balanced ternary representation), A030102, A134021, A274107.

A134027 gives the numbers whose balanced ternary representation is palindromic.

Programs

Python

def a(n):
    if n==0: return 0
    s=[]
    x=0
    while n>0:
        x=n%3
        n=n//3
        if x==2:
            x=-1
            n+=1
        s.append(x)
    l=s[::-1]
    return sum(l[i]*3**i for i in range(len(l)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 10 2017

Formula

a(A134027(n)) = A134027(n);

A134021(ABS(a(n))) <= A134021(n).