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

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

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

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

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

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

Solving
0 = 0

Couldn't find a variable to solve for.

This equation is an identity, all real numbers are solutions.