cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A330167 Length of the longest run of 1's in the ternary expression of n.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1, 1, 2, 3, 2, 1, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 3, 4, 3, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1, 1, 2, 3, 2, 1, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Joshua Oliver, Dec 04 2019

Keywords

Comments

All numbers appear in this sequence. Numbers of the form (3^n-1)/2 (A003462(n)) have n 1's in their ternary expression.
The longest run of zeros possible in this sequence is 2, as the last digit of the ternary expression of the integers cycles between 0, 1, and 2, meaning that at least one of three consecutive numbers has a 1 in its ternary expression.

Examples

			For n = 43, the ternary expression of 43 is 1121. The length of the runs of 1's in the ternary expression of 43 are 2 and 1, respectively. The larger of these two values is 2, so a(43) = 2.
   n [ternary n] a(n)
   0 [        0] 0
   1 [        1] 1
   2 [        2] 0
   3 [      1 0] 1
   4 [      1 1] 2
   5 [      1 2] 1
   6 [      2 0] 0
   7 [      2 1] 1
   8 [      2 2] 0
   9 [    1 0 0] 1
  10 [    1 0 1] 1
  11 [    1 0 2] 1
  12 [    1 1 0] 2
  13 [    1 1 1] 3
  14 [    1 1 2] 2
  15 [    1 2 0] 1
  16 [    1 2 1] 1
  17 [    1 2 2] 1
  18 [    2 0 0] 0
  19 [    2 0 1] 1
  20 [    2 0 2] 0

Links

Crossrefs

Cf. A003462, A007089, A062756, A330036, A330166, A330168.
Equals zero iff n is in A005823.

Programs

  • Mathematica
    Table[Max@FoldList[If[#2==1,#1+1,0]&,0,IntegerDigits[n,3]],{n,0,90}]
Table[Max[Length/@Select[Split[IntegerDigits[n,3]],MemberQ[#,1]&]],{n,0,100}]/.(-\[Infinity]->0) (* Harvey P. Dale, Jan 07 2023 *)

Formula

a(A003462(n)) = a((3^n-1)/2) = n.
a(n) = 0 iff n is in A005823.

A330168 Length of the longest run of 2's in the ternary expression of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0
Offset: 0

Views

Author

Joshua Oliver, Dec 04 2019

Keywords

Comments

All numbers appear in this sequence. Numbers of the form 3^n-1 (A024023(n)) have n 2's in their ternary expression.
The longest run of zeros possible in this sequence is 2, as the last digit of the ternary expression of the integers cycles between 0, 1, and 2, meaning that at least one of three consecutive numbers has a 2 in its ternary expression.

Examples

			For n = 74, the ternary expression of 74 is 2202. The length of the runs of 2's in the ternary expression of 74 are 2 and 1, respectively. The larger of these two values is 2, so a(74) = 2.
   n [ternary n] a(n)
   0 [        0] 0
   1 [        1] 0
   2 [        2] 1
   3 [      1 0] 0
   4 [      1 1] 0
   5 [      1 2] 1
   6 [      2 0] 1
   7 [      2 1] 1
   8 [      2 2] 2
   9 [    1 0 0] 0
  10 [    1 0 1] 0
  11 [    1 0 2] 1
  12 [    1 1 0] 0
  13 [    1 1 1] 0
  14 [    1 1 2] 1
  15 [    1 2 0] 1
  16 [    1 2 1] 1
  17 [    1 2 2] 2
  18 [    2 0 0] 1
  19 [    2 0 1] 1
  20 [    2 0 2] 1

Links

Crossrefs

Cf. A007089, A024023, A081603, A330036, A330166, A330167.
Equals zero iff n is in A005836.

Programs

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

Formula

a(A024023(n)) = a(3^n-1) = n.
a(n) = 0 iff n is in A005836.

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

Original entry on oeis.org

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

Views

Author

Joshua Oliver, Feb 05 2020

Keywords

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.

Examples

			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

Links

Crossrefs

Cf. A003462, A007089, A062756, A330036.
Equals A330166 + A330167 + A330168.

Programs

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

Formula

a(n) = A330166(n) + A330167(n) + A330168(n).
a(A003462(n)) = a(A024023(n)) = n.
Showing 1-3 of 3 results.

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