MathDB

For each positive integer

k

, define

r(k)

as the number of runs of

k

in base-

2

, where a run is a collection of consecutive

0

s or consecutive

1

s without a larger one containing it. For example,

(11100100)_2

has

4

runs, namely

111-00-1-00

. Also,

r(0) = 0

. Given a positive integer

n

, find all functions

f : \mathbb{Z} \rightarrow\mathbb{Z}

such that

\sum_{k=0}^{2^n-1} 2^{r(k)}f(k+(-1)^{k} x)=(-1)^{x+n}\text{ for all integer $x$.}

Proposed by YaWNeeT

N