A(n)=A(n-1)+A(n-2)+A(n-5)

Solution for A(n)=A(n-1)+A(n-2)+A(n-5) equation:

Simplifying
A(n) = A(n + -1) + A(n + -2) + A(n + -5)

Multiply A * n
nA = A(n + -1) + A(n + -2) + A(n + -5)

Reorder the terms:
nA = A(-1 + n) + A(n + -2) + A(n + -5)
nA = (-1 * A + n * A) + A(n + -2) + A(n + -5)
nA = (-1A + nA) + A(n + -2) + A(n + -5)

Reorder the terms:
nA = -1A + nA + A(-2 + n) + A(n + -5)
nA = -1A + nA + (-2 * A + n * A) + A(n + -5)
nA = -1A + nA + (-2A + nA) + A(n + -5)

Reorder the terms:
nA = -1A + nA + -2A + nA + A(-5 + n)
nA = -1A + nA + -2A + nA + (-5 * A + n * A)
nA = -1A + nA + -2A + nA + (-5A + nA)

Reorder the terms:
nA = -1A + -2A + -5A + nA + nA + nA

Combine like terms: -1A + -2A = -3A
nA = -3A + -5A + nA + nA + nA

Combine like terms: -3A + -5A = -8A
nA = -8A + nA + nA + nA

Combine like terms: nA + nA = 2nA
nA = -8A + 2nA + nA

Combine like terms: 2nA + nA = 3nA
nA = -8A + 3nA

Solving
nA = -8A + 3nA

Solving for variable 'n'.

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

Add '-3nA' to each side of the equation.
nA + -3nA = -8A + 3nA + -3nA

Combine like terms: nA + -3nA = -2nA
-2nA = -8A + 3nA + -3nA

Combine like terms: 3nA + -3nA = 0
-2nA = -8A + 0
-2nA = -8A

Divide each side by '-2A'.
n = 4

Simplifying
n = 4