Core idea
1. What changes when you move p, q, or the seeds?
The coefficients p and q change the entire rule; the seeds select one particular orbit within that rule. Two users can choose the same p and q and still obtain very different sequences if they start from different seeds.
Fibonacci is the case p = 1, q = 1, a₀ = 0, and a₁ = 1: 0, 1, 1, 2, 3, 5… Changing only the seeds preserves the rule “add the previous two,” but it no longer necessarily produces the classical Fibonacci sequence.
- Change p or q: you change the modes the recurrence can contain.
- Change a₀ or a₁: you activate, combine, or even cancel those modes.
Reading the results
2. Two different questions: is the rule stable, and what does this orbit do?
Characteristic roots describe the rule. Roughly, a root with modulus below 1 tends to fade, one above 1 tends to grow, and one on the unit circle marks a boundary case.
Seeds still matter because they may cancel an expanding mode. The calculator therefore separates spectral stability —a property of the rule— from orbit behavior —what happens with your seeds—.
- Evolution: each orange point is a term at an integer index; the line only helps you follow their order.
- Consecutive ratio: shows aₙ/aₙ₋₁ and can reveal a dominant mode.
- A gap in the line: means aₙ₋₁ = 0, so that ratio is undefined.
Keep in mind: A ratio that appears to settle is numerical evidence, not a universal guarantee. Complex roots, tied moduli, or zero denominators may prevent convergence.
Try it in the calculator
3. Three short experiments showing different behaviors
- 1Fibonacci: use p=1, q=1, a₀=0, a₁=1, and N=25. In Consecutive Ratio, enable Convergence Focus and watch the values approach φ≈1.618.
- 2Polynomial growth: use p=2, q=−1, a₀=1, and a₁=2. You get 1,2,3,4…: a repeated root r=1 does not grow exponentially, but its n factor still grows.
- 3Cancelled expanding mode: use p=3, q=−2, a₀=1, and a₁=1. The rule is spectrally unstable, although this particular orbit remains constant.