Recurrence Relation Calculator
Solve linear recurrence relations, find characteristic roots, and closed-form solutions.
Recurrence Type
a_n = 1a_{n-1} + 1a_{n-2}
Sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
Closed Form
a_n = 0.4472×(1.6180)^n + -0.4472×(-0.6180)^n
Characteristic Roots
r₁ = 1.618034
r₂ = -0.618034
Discriminant: 5.0000
Ratio Limit
lim(aₙ₊₁/aₙ) ≈ 1.6180257511
Full Sequence
a_0=0a_1=1a_2=1a_3=2a_4=3a_5=5a_6=8a_7=13a_8=21a_9=34a_10=55a_11=89a_12=144a_13=233a_14=377
Recurrence Relations
Solving Method
- Find characteristic equation: r² - c₁r - c₂ = 0
- Solve for roots r₁, r₂
- General solution: aₙ = Ar₁ⁿ + Br₂ⁿ
- Use initial conditions to find A, B
Special Cases
- Repeated roots: aₙ = (A + Bn)rⁿ
- Complex roots: involves trig functions
- Fibonacci: φⁿ/√5 approximation