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

Solution for f(2n+1)=f(n)+f(n-1) equation:

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

Reorder the terms:
f(1 + 2n) = f(n) + f(n + -1)
(1 * f + 2n * f) = f(n) + f(n + -1)
(1f + 2fn) = f(n) + f(n + -1)

Multiply f * n
1f + 2fn = fn + f(n + -1)

Reorder the terms:
1f + 2fn = fn + f(-1 + n)
1f + 2fn = fn + (-1 * f + n * f)
1f + 2fn = fn + (-1f + fn)

Reorder the terms:
1f + 2fn = -1f + fn + fn

Combine like terms: fn + fn = 2fn
1f + 2fn = -1f + 2fn

Add '-2fn' to each side of the equation.
1f + 2fn + -2fn = -1f + 2fn + -2fn

Combine like terms: 2fn + -2fn = 0
1f + 0 = -1f + 2fn + -2fn
1f = -1f + 2fn + -2fn

Combine like terms: 2fn + -2fn = 0
1f = -1f + 0
1f = -1f

Solving
1f = -1f

Solving for variable 'f'.

Move all terms containing f to the left, all other terms to the right.

Add 'f' to each side of the equation.
1f + f = -1f + f

Combine like terms: 1f + f = 2f
2f = -1f + f

Combine like terms: -1f + f = 0
2f = 0

Divide each side by '2'.
f = 0

Simplifying
f = 0