A362756 Sum of the bits of the "fractional part" of the base-phi representation of n.

0, 0, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 4, 3, 3, 3, 3, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 4, 3, 3, 3, 3, 2, 2, 2, 3, 3, 3, 4, 3, 3, 3, 4, 4, 4, 5, 4, 4, 4, 4, 3, 3, 3, 4, 4

Offset

0,6

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(-1).

The first difference of a(n) is Fibonacci-automatic and takes values in {-1,0,1} only.

Links

Table of n, a(n) for n=0..86.

George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), 98-110.

Formula

There is a linear representation of rank 21 for a(n).

Example

For n = 20 the phi-representation is 1000010.010001, so a(20) = 2.

Crossrefs

Cf. A055778, A362716.

Sequence in context: A161093 A282904 A299825 * A289493 A324341 A271325

Adjacent sequences: A362753 A362754 A362755 * A362757 A362758 A362759

Keyword

nonn

Author

Jeffrey Shallit, May 02 2023

Status

approved