A332025 Sum of the lengths of the longest runs of 0, 1, and 2 in the ternary expression of n.

1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 4, 3, 4, 2, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 3, 4, 4, 3, 3, 3, 3, 2, 4, 3, 4, 4, 4, 3, 3, 4, 3, 2, 3, 3, 4, 3, 3, 4, 4, 3, 3, 2, 3, 4, 4, 3, 4, 4, 3, 4, 4, 4, 5, 4, 5, 3, 4, 4, 4, 4, 5, 3

Offset

0,4

Comments

All positive integers appear in this sequence. Given some number k, there will always be some ternary number that has k 1's or k 2's.

The number 0 never appears in this sequence, as every number has at least 1 digit.

Links

Table of n, a(n) for n=0..90.

Wikipedia, Ternary numeral system.

Formula

a(n) = A330166(n) + A330167(n) + A330168(n).

a(A003462(n)) = a(A024023(n)) = n.

Example

For n = 268, the ternary expansion of 268 is 100221. The length of the run of 0's in the ternary expansion of 268 is 2. The length of the runs of 1's in the ternary expansion of 268 are 1 and 1 respectively. The length of the run of 2's in the ternary expansion of 268 is 2. The sum of 2, 1, and 2 is 5, so a(268) = 5.
   n [ternary n] A330166(n) + A330167(n) + A330168(n) = a(n)
   0 [        0] 1          + 0          + 0          = 1
   1 [        1] 0          + 1          + 0          = 1
   2 [        2] 0          + 0          + 1          = 1
   3 [      1 0] 1          + 1          + 0          = 2
   4 [      1 1] 0          + 2          + 0          = 2
   5 [      1 2] 0          + 1          + 1          = 2
   6 [      2 0] 1          + 0          + 1          = 2
   7 [      2 1] 0          + 1          + 1          = 2
   8 [      2 2] 0          + 0          + 2          = 2
   9 [    1 0 0] 2          + 1          + 0          = 3
  10 [    1 0 1] 1          + 1          + 0          = 2
  11 [    1 0 2] 1          + 1          + 1          = 3
  12 [    1 1 0] 1          + 2          + 0          = 3
  13 [    1 1 1] 0          + 3          + 0          = 3
  14 [    1 1 2] 0          + 2          + 1          = 3
  15 [    1 2 0] 1          + 1          + 1          = 3
  16 [    1 2 1] 0          + 1          + 1          = 2
  17 [    1 2 2] 0          + 1          + 2          = 3
  18 [    2 0 0] 2          + 0          + 1          = 3
  19 [    2 0 1] 1          + 1          + 1          = 3
  20 [    2 0 2] 1          + 0          + 1          = 2

Mathematica

Table[Sum[Max@FoldList[If[#2==k, #1+1, 0]&, 0, IntegerDigits[n, 3]], {k, 0, 2}], {n, 1, 90}]

Crossrefs

Cf. A003462, A007089, A062756, A330036.

Equals A330166 + A330167 + A330168.

Sequence in context: A081326 A277879 A257071 * A212300 A165074 A374152

Adjacent sequences: A332022 A332023 A332024 * A332026 A332027 A332028

Keyword

nonn,base

Author

Joshua Oliver, Feb 05 2020

Status

approved