A358915 a(n) is the far-difference representation of n written in balanced ternary.

0, 1, 3, 9, 26, 27, 78, 80, 81, 82, 234, 240, 242, 243, 244, 246, 702, 703, 720, 726, 728, 729, 730, 732, 738, 2105, 2106, 2107, 2109, 2160, 2161, 2178, 2184, 2186, 2187, 2188, 2190, 2196, 2213, 2214, 6315, 6317, 6318, 6319, 6321, 6327, 6479, 6480, 6481, 6483

Offset

0,3

Comments

A far-difference representation of an integer is the unique way to write that integer of the sum/difference of Fibonacci numbers such that any two terms in the sum with the same sign differ by at least an index of 4 and any two terms with different signs differ by an index of at least 3.

This sequence is also the list of numbers whose balanced ternary representation has the property that all signed-digits with the same sign differ by at least 4 positions and all signed-digits with different signs differ by at least 3 positions.

Links

Table of n, a(n) for n=0..49.

Hannah Alpert, Differences of Multiple Fibonacci Numbers, Integers, 9 (2009), 745-749.

Example

Let F_i be the i-th term of the 0-indexed Fibonacci sequence beginning 1, 2, 3, 5, 8, ... .
| n  | far-difference               | a(n)                  |
|----+------------+-----------------+-----------------+-----+
| 10 | 13 - 3     | F_5 - F_2       | 3^5 - 3^2       | 234 |
| 11 | 13 - 2     | F_5 - F_1       | 3^5 - 3^1       | 240 |
| 12 | 13 - 1     | F_5 - F_0       | 3^5 - 3^0       | 242 |
| 13 | 13         | F_5             | 3^5             | 243 |
| 14 | 13 + 1     | F_5 + F_0       | 3^5 + 3^0       | 244 |
| 15 | 13 + 2     | F_5 + F_1       | 3^5 + 3^1       | 246 |
| 16 | 21 - 5     | F_6 - F_3       | 3^6 - 3^3       | 702 |
| 17 | 21 - 5 + 1 | F_6 - F_3 + F_0 | 3^6 - 3^3 + 3^0 | 703 |
| 18 | 21 - 3     | F_6 - F_2       | 3^6 - 3^2       | 720 |
| 19 | 21 - 2     | F_6 - F_1       | 3^6 - 3^1       | 726 |
| 20 | 21 - 1     | F_6 - F_0       | 3^6 - 3^0       | 728 |

Crossrefs

Cf. A003714, A097083, A105446, A117966.

Sequence in context: A119825 A235538 A370642 * A218916 A037260 A035313

Adjacent sequences: A358912 A358913 A358914 * A358916 A358917 A358918

Keyword

nonn,base

Author

Peter Kagey, Dec 05 2022

Status

approved