A362716 Sum of the bits of the "integer part" of the base-phi representation of n.

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

Offset: 0

Author

Jeffrey Shallit, Apr 30 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 "integer" part is the string of bits b(R)b(R-1)...b(1)b(0).

Examples

			For n = 20 we have n = phi^6 + phi^1 + phi^(-2) + phi^(-6), and the "integer part" has 2 terms, so a(20) = 2.
		

Links

• George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), 98-110.
• Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.

Crossrefs

Cf. A055778.

Formula

There is a linear representation of rank 19 for a(n).