A309119 a(n) is the number of 1's minus the number of 2's among the ternary representations of the integers in the interval [0..n].

0, 1, 0, 1, 3, 3, 2, 2, 0, 1, 3, 3, 5, 8, 9, 9, 10, 9, 8, 8, 6, 6, 7, 6, 4, 3, 0, 1, 3, 3, 5, 8, 9, 9, 10, 9, 11, 14, 15, 18, 22, 24, 25, 27, 27, 27, 28, 27, 28, 30, 30, 29, 29, 27, 26, 26, 24, 24, 25, 24, 22, 21, 18, 18, 19, 18, 19, 21, 21, 20, 20, 18, 16, 15

Offset: 0

Author

Rémy Sigrist, Jul 13 2019

Keywords

• nonn
• look
• base

Comments

This sequence has connections with a Takagi (or blancmange) curve.

Let t be the real function defined over [0..1] as follows:

- t(x) = 0 for x in [0..1/3],

- t(x) = x - 1/3 for x in ]1/3..2/3],

- t(x) = 1 - x for x in ]2/3..1].

Let g be the real function defined over [0..1] as follows:

- g(x) = Sum_{k >= 0} t(x * 3^k)/3^k.

The representation of n -> (n/3^k, a(n)/3^k) for n = 0..3^k converges to the representation of g over [0..1] as k tends to infinity.

Examples

			The first terms, alongside the ternary expansion of n and the corresponding number of 1's and 2's, are:
  n   a(n)  ter(n)  A062756(n)  A081603(n)
  --  ----  ------  ----------  ----------
   0     0       0           0           0
   1     1       1           1           0
   2     0       2           0           1
   3     1      10           1           0
   4     3      11           2           0
   5     3      12           1           1
   6     2      20           0           1
   7     2      21           1           1
   8     0      22           0           2
   9     1     100           1           0
  10     3     101           2           0
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..6560
• Rémy Sigrist, Colored pinplot of the sequence for n = 0..3^7-1 (where the color denotes the contribution of the digits according to their position in the ternary expansion)
• Wikipedia, Blancmange curve

Crossrefs

Cf. A024023, A062756, A081603.

Programs

Mathematica

Accumulate[Table[Total[IntegerDigits[n,3]/.(2->-1)],{n,0,80}]] (* Harvey P. Dale, Jun 23 2020 *)

PARI

s = 0; for (n=0, 73, t = digits(n,3); print1 (s+=sum(i=1, #t, if (t[i]==1, +1, t[i]==2, -1, 0)) ", "))

Formula

a(n) = Sum_{k = 0..n} (A062756(k) - A081603(k)).

a(n) >= 0 with equality iff n = 3^k - 1 for some k >= 0 (A024023).

a(3*k + 2) = 3*a(k) for any k >= 0.

a(3^k + m) = a(m) + m + 1 for any k >= 0 and m = 0..3^k-1.

a(2*3^k + m) = a(m) + 3^k - m - 1 for any k >= 0 and m = 0..3^k-1.