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A362872
Length of the "fractional part" of the phi-representation of n.
1
0, 0, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10
OFFSET
0,3
COMMENTS
The phi-representation of n is the (essentially) unique way to write n = Sum_{j=L..R} b(j)*phi^j, where b(j) is in {0,1} and -oo < L <= 0 <= R, where phi = (1+sqrt(5))/2, subject to the condition that b(j)b(j+1) != 1. The "fractional" part is the string of bits b(L)b(L+1)...b(-1), and its length is thus L.
The gaps between consecutive terms are all either 0 or 2, and a gap of 2 occurs if and only if n = L(2i+1) for i >= 0. This is equivalent to Theorem 2.1 of Sanchis and Sanchis (2001).
LINKS
George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), 98-110.
G. R. Sanchis and L. A. Sanchis, On the frequency of occurrence of α^i in the α-expansions of the positive integers, Fibonacci Quart. 39 (2001), 123-137.
FORMULA
There is a linear representation of rank 11 for a(n).
EXAMPLE
The phi-representation of 20 is 1000010.010001, so a(20) = 6.
CROSSREFS
Sequence in context: A219654 A096491 A217871 * A306390 A106160 A305117
KEYWORD
nonn
AUTHOR
Jeffrey Shallit, May 07 2023
STATUS
approved