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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]. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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.

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

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.

EXAMPLE

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

PROG

(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)) ", "))

CROSSREFS

Cf. A024023, A062756, A081603.

Sequence in context: A102905 A020862 A131589 * A308725 A202691 A328143

Adjacent sequences:  A309116 A309117 A309118 * A309120 A309121 A309122

KEYWORD

nonn,look,base

AUTHOR

Rémy Sigrist, Jul 13 2019

STATUS

approved

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Last modified June 6 04:15 EDT 2020. Contains 334859 sequences. (Running on oeis4.)