# Recurrence relation
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## Microsims — p5.js
### Recurrence relation (p5.js) · `y[n]=f(y[n−1])`
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<iframe src="https://editor.p5js.org/sciencenibber/full/QPai0xNyx" width="100%" height="480" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Recurrence relation — p5.js microsim"></iframe>
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*Define each value from previous ones — the discrete-time equation behind IIR filters and dynamic systems.*
**Open in the editor:** [▶ fork this sketch](https://editor.p5js.org/sciencenibber/sketches/QPai0xNyx) · movement *VI · Control theory & estimation* · library `p5js`
### Related microsims
Live sims on neighbouring articles — 3 of them inside this article's own Wikipedia link tree:
- [[Chaos_theory]] *(in tree)*
- [[Digital_signal_processing]] *(in tree)*
- [[Z-transform]] *(in tree)*
- [[Control_theory]]
- [[Cruise_control]]
- [[Detection_theory]]
*Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.*
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## Links (Wikipedia order)
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`Abramov's_algorithm` · `Absolute_value` · [[Algorithm]] · `Analysis_of_algorithms` · `Bessel_function` · `Binomial_coefficient` · [[Chaos_theory]] · `Characteristic_polynomial` · `Charles_E._Leiserson` · `Clifford_Stein` · `Closed-form_expression` · `Comb_filter` · `Combinatorial_principles` · [[Computer_science]] · `Continued_fraction` · `Derivative` · `Difference_Equations:_From_Rabbits_to_Chaos` · `Digital_filter` · [[Digital_signal_processing]] · `Discretization` · `Divide-and-conquer_algorithm` · `Donald_Knuth` · `Dyadic_transformation` · [[Ecology]] · `Edward_C._Prescott` · `Encyclopedia_of_Mathematics` · `Equation` · `Eric_W._Weisstein` · `Factorial` · `Finite_difference` · `Function_(mathematics)` · `Generating_function` · `Holonomic_function` · `Infinite_impulse_response` · `Initial_condition` · `Initial_value_problem` · `Integral_equation` · `Integration_by_reduction_formulae` · `Integrodifference_equation` · `Interest_rate` · `Introduction_to_Algorithms` · `Inverse_function` · `Iterated_function` · `Lagged_Fibonacci_generator` · `Lars_Ljungqvist` · `Limit_of_a_sequence` · `Linear_differential_equation` · `Linear_function` · `Linear_recurrence_with_constant_coefficients` · `Logistic_map` · `Mary_Flahive` · `Master_theorem_(analysis_of_algorithms)` · `MathWorld` · `Mathematical_induction` · `Mathematics` · `Matrix_difference_equation` · `Nancy_Stokey` · `Nathan_Jacobson` · `Natural_number` · `Nicholson–Bailey_model` · `Operator_(mathematics)` · [[Ordinary_differential_equation]] · `Orthogonal_polynomials` · `P-recursive_equation` · `Pascal's_triangle` · `Petkovšek's_algorithm` · `Polynomial_solutions_of_P-recursive_equations` · `Population_(biology)` · [[Population_dynamics]] · `Rational_difference_equation` · `Rational_function` · `Recursion` · [[Recursion_(computer_science)]] · `Ronald_Graham` · [[Sequence]] · `Shift_operator` · `Simultaneous_equations_model` · `Special_functions` · `Stability_theory` · `Summation_equation` · `Taylor_series` · `Tent_map` · `Thomas_H._Cormen` · `Thomas_J._Sargent` · `Time_complexity` · `Tuple` · `Voltinism` · [[Z-transform]]
> Signal Processing concept · part of the Signal Processing Portal · movement VI · !01 制御 seigyo.svg
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*Rendered from the live microsim (▶ motion).*
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## See it next
[](Dynamic_system)
*→ Dynamic system*
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Back to Signal Processing Portal · the room · Semiotic gateway
## What it is
A recurrence relation is an equation that defines each term of a sequence as a function of one or more preceding terms, so the sequence is generated step by step from initial values.
## How it works / why it matters
In discrete-time signal processing the recurrence $y[n] = \sum b_k x[n-k] - \sum a_k y[n-k]$ is exactly the difference equation of an IIR filter or a discrete dynamic system: past outputs feed back into the present output. Whether such a system is stable depends on the roots of its characteristic equation — inside the unit circle for stability — and its closed-form behavior can be found by z-transform or by solving that characteristic equation. Recurrences describe everything from digital filters and autoregressive models to iterative algorithms.
## Signs & universals
Instantiates the semiotic universals: signal · feedback · stability.
## Related
The discrete-time engine of a Dynamic system and the equation form of a discrete [[Linear_time-invariant_system]]; it defines the Autoregressive model and its z-domain [[Transfer_function]].
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*Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].*
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## Wikipedia : Wikitube
**Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Recurrence_relation) : [Wikitube](https://en.wikitube.io/wiki/Recurrence_relation)
## Previous hub tags
Tree parents: [[Dynamical_system]] · [[Phase_space]] · [[Stock_and_flow]].
Legacy hubs: none.
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*Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*