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

Author

Jeffrey Shallit, May 02 2023

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

Cf. A055778, A362716.

Formula

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