A309119 - OEIS

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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].

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

MATHEMATICA

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

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: A020862 A131589 A338113 * A308725 A202691 A374410

Adjacent sequences: A309116 A309117 A309118 * A309120 A309121 A309122

KEYWORD

nonn,look,base

AUTHOR

Rémy Sigrist, Jul 13 2019

STATUS

approved