OFFSET
1
COMMENTS
According to Theorem 4 of Leung (2023), the golden ratio (or phi) base expansion of any rational number 0 < p/q < 1, where p and q are positive integers and gcd(p,q) = 1, is strictly periodic with period A001175(q).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..7968 (first 1000 rows)
King Shun Leung, phi-Expansions of Rationals, The Fibonacci Quarterly, Vol. 61, No. 2 (2023), pp. 162-166.
EXAMPLE
row 1 is {0} since the golden ratio base representation of 1/1 is 1.000...
row 2 is {0, 1, 0} since the golden ratio base representation of 1/2 is 0.(010)(010)(010)... (1/2 = 1/phi^2 + 1/phi^5 + 1/phi^8 + ..., where phi is the golden ratio, A001622).
row 3 is {0, 0, 1, 0, 1, 0, 0, 0} since the golden ratio base representation of 1/3 is 0.(00101000)(00101000)(00101000)...
The first 9 rows are:
n | row n
---+------------------------------------------------------------------------
1 | 0
2 | 0, 1, 0
3 | 0, 0, 1, 0, 1, 0, 0, 0
4 | 0, 0, 1, 0, 0, 0
5 | 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0
6 | 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0
7 | 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0
8 | 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0
9 | 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0
MATHEMATICA
CROSSREFS
KEYWORD
nonn,tabf,easy,base
AUTHOR
Amiram Eldar, Mar 17 2025
STATUS
approved