Let-f-define-such-as-f-1-1-f-3-3-n-2-f-2n-f-n-f-4n-1-2f-2n-1-f-n-f-4n-3-3f-2n-1-2f-n-1-Prove-that-n-f-n-is-odd-2-Prove-that-if-f-a-n-a-n-then-a-n-2-n-1-or-a-n-2-n-1

Question Number 139323 by snipers237 last updated on 25/Apr/21

L e t f d e f i n e s u c h a s f (1) = 1, f (3) = 3

\forall n ⩾ 2, f (2 n) = f (n)

f (4 n + 1) = 2 f (2 n + 1) - f (n)

f (4 n + 3) = 3 f (2 n + 1) - 2 f (n)

1) P r o v e t h a t \forall n, f (n) i s o d d

2) P r o v e t h a t i f f (a_{n}) = a_{n},

t h e n a_{n} = 2^{n} - 1 o r a_{n} = 2^{n} + 1