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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
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, 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
KEYWORD
nonn,look,base
AUTHOR
Rémy Sigrist, Jul 13 2019
STATUS
approved