Problem 5

Let

\mathbb{N}=\{1,2,3,\ldots\}

. Determine whether there exists a strictly increasing function

f:\mathbb{N}\mapsto\mathbb{N}

such that (i)

f(1)=2

; (ii)

f(f(n))=f(n)+n

(n\in\mathbb{N})

Step 2 of 5: Check the initial value

In plain words

The shift by phi minus 1 changes the usual golden-ratio floor sequence so that it begins at 2.

f(1)=\lfloor2\varphi-1\rfloor=2

Detailed analysis

Using the defining quadratic equation for phi, the value at 1 lies between 2 and 3, so its floor is 2.