Recurrence Relations Calculator

Linear Recurrence Relations

A linear recurrence relation with constant coefficients has the form: a_n = c_1a_n-1 + c_2a_n-2 + ·s + c_ka_n-k + f(n) where c_1, c_2, ..., c_k are constants and f(n) is a function of n (if f(n)=0, the recurrence is homogeneous). The order is k (how many previous terms appear). Initial conditions a_0, a_1, ..., a_k-1 determine a unique sequence. Examples include: Fibonacci (k=2, c_1=c_2=1, f=0), geometric sequences (k=1), and many algorithm analyses. The linearity and constant coefficients make these recurrences solvable using the characteristic equation method, providing explicit closed-form solutions.

a_n = c_1a_n-1 + c_2a_n-2 + ·s + c_ka_n-k + f(n)

Characteristic Equation Method

For homogeneous linear recurrences, assume solutions of the form a_n = r^n. Substituting into a_n = c_1a_n-1 + c_2a_n-2 + ·s + c_ka_n-k gives: r^n = c_1r^n-1 + c_2r^n-2 + ·s + c_kr^n-k Dividing by r^n-k yields the characteristic equation: r^k - c_1r^k-1 - c_2r^k-2 - ·s - c_k = 0 For distinct roots r_1, r_2, ..., r_k, the general solution is: a_n = A_1r_1^n + A_2r_2^n + ·s + A_kr_k^n where constants A_i are determined by initial conditions. For Fibonacci, the characteristic equation r^2 - r - 1 = 0 has roots \phi = (1+√(5))/2 (golden ratio) and \psi = (1-√(5))/2, giving Binet's formula.

r^k - c_1r^k-1 - c_2r^k-2 - ·s - c_k = 0

Repeated Roots and Special Cases

When the characteristic equation has repeated roots, the solution form changes. If r is a root of multiplicity m, it contributes terms: A_1r^n + A_2nr^n + A_3n^2r^n + ·s + A_mn^m-1r^n For example, a_n = 2a_n-1 - a_n-2 has characteristic equation (r-1)^2 = 0 with double root r=1, giving solution a_n = A + Bn. For complex roots r = α ± β i = \rho e^± iθ (in polar form), the solution can be written as: a_n = \rho^n[A\cos(nθ) + B\sin(nθ)] This form reveals oscillatory behavior. Complex roots arise from recurrences modeling oscillations, such as discretized wave equations.

a_n = \rho^n[A\cos(nθ) + B\sin(nθ)] \quad \text(complex roots)

Non-Homogeneous Recurrences

For non-homogeneous recurrences a_n = c_1a_n-1 + ·s + c_ka_n-k + f(n), the solution is: a_n = a_n^(h) + a_n^(p) where a_n^(h) is the general solution to the homogeneous equation and a_n^(p) is any particular solution. Finding particular solutions uses methods similar to ODEs: if f(n) is polynomial, try polynomial form; if exponential r^n, try Ar^n (unless r is a characteristic root, then try Anr^n). For example, a_n = a_n-1 + n has homogeneous solution A and particular solution n(n+1)/2, giving a_n = A + n(n+1)/2. This method systematically handles recurrences with non-zero right-hand sides.

a_n = a_n^(h) + a_n^(p)

Generating Function Method

Generating functions provide an alternative, often more powerful approach. Define G(x) = Σ_n=0^∞ a_n x^n. Multiply the recurrence relation by x^n and sum over all valid n. Use initial conditions and algebra to derive an equation for G(x). Solve for G(x), then extract coefficients. For Fibonacci F_n = F_n-1 + F_n-2 with F_0=0, F_1=1: G(x) = (x)/(1-x-x^2) Partial fraction decomposition gives: G(x) = (1)/(√(5))((1)/(1-\phi x) - (1)/(1-\psi x)) Expanding yields Binet's formula directly. This method handles non-constant coefficients and more complex recurrences than the characteristic equation approach.

G(x) = Σ_n=0^∞ a_n x^n

Master Theorem for Divide-and-Conquer

Many algorithms satisfy recurrences of the form T(n) = aT(n/b) + f(n) where a subproblems of size n/b each are solved. The Master Theorem provides immediate solutions: Let c = \log_b a. Case 1: If f(n) = O(n^d) with d < c, then T(n) = \Theta(n^c). Case 2: If f(n) = \Theta(n^c), then T(n) = \Theta(n^c \log n). Case 3: If f(n) = \Omega(n^d) with d > c and regularity condition, then T(n) = \Theta(f(n)). Examples: Merge sort T(n) = 2T(n/2) + n has a=2, b=2, c=1, d=1 (Case 2), so T(n) = \Theta(n\log n). Binary search T(n) = T(n/2) + 1 has c=0, d=0 (Case 2), giving T(n) = \Theta(\log n).

T(n) = aT(n/b) + f(n) \Rightarrow T(n) = \Theta(n^\log_b a\log n) \text (Case 2)

Applications Across Mathematics and Science

Recurrence relations model discrete-time dynamical systems throughout science. In population biology, the logistic map x_n+1 = rx_n(1-x_n) exhibits chaotic behavior for certain r. In economics, difference equations model discrete-time markets and fiscal policy effects. In physics, discretized differential equations become recurrence relations (finite difference methods). Computer science algorithms are rife with recurrences: quicksort, merge sort, binary search, dynamic programming all produce recurrence relations for runtime or space complexity. Number theory uses recurrences to generate sequences (Pascal's triangle, Catalan numbers). Understanding recurrence relations provides insight into long-term behavior, stability, and asymptotic growth rates of discrete processes.

\textApplications: algorithms, population dynamics, numerical methods, combinatorics

Worked Example

Inputs

• recurrenceType: fibonacci
• a0: 0
• a1: 1
• nthTerm: 10

Result: a₁₀ = 55

Explanation

Fibonacci sequence: F₁₀ = 55 using Fₙ = Fₙ₋₁ + Fₙ₋₂ with F₀=0, F₁=1

Linear Recurrence Order 2

Inputs

• recurrenceType: linear2
• a0: 1
• a1: 2
• nthTerm: 5

Result: a₅ = 32

Explanation

Recurrence aₙ = 3aₙ₋₁ - 2aₙ₋₂ with a₀=1, a₁=2 gives a₅ = 32