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

0,5

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).

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.

Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.

Formula

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

Example

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.

Crossrefs

Cf. A055778.

Sequence in context: A025877 A219182 A184171 * A133989 A379375 A029398

Adjacent sequences: A362713 A362714 A362715 * A362717 A362718 A362719

Keyword

nonn

Author

Jeffrey Shallit, Apr 30 2023

Status

approved