happytails.ai

Recurrence relation explorer

Open Recurrence relation explorer on Happytails.ai. This view has the same page content and links, with explicit tool capabilities. Read the processing notes before using a tool.

Tool capabilities

{
  "id": "recurrence-relation-explorer",
  "category": "calculators",
  "built": true,
  "requirements": "Inputs are limited to 1,000 characters, order 10, forcing polynomials of degree 3 and 1 to 200 terms; exact values may grow to 1,024 digits. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of terms.",
  "implementation": {
    "fields": [
      {
        "key": "terms",
        "label": "Number of terms",
        "value": "20",
        "type": "number",
        "min": 1,
        "max": 200,
        "step": 1,
        "help": "How many terms to list, from 1 to 200, including the initial values."
      }
    ],
    "presentation": "metrics",
    "inputCodeDirection": "ltr",
    "registrationFamily": "exact-math-discrete",
    "inputMaxLength": 1000,
    "inputErrorCode": "EXACT_MATH_DISCRETE_EXPRESSION_TOO_LONG",
    "note": "Expands the recurrence exactly on your device and solves linear recurrences with constant coefficients in closed form, up to order 10.",
    "experience": {
      "automatic": false,
      "single": false,
      "inputMaxLength": 1000,
      "inputMode": "text",
      "label": "Recurrence and initial values",
      "placeholder": "a(n) = a(n-1) + a(n-2), a(0) = 0, a(1) = 1",
      "help": "Write the recurrence first, then initial values separated by commas or new lines. Up to 1,000 characters. Review the settings before running.",
      "helpParts": [
        "Write the recurrence first, then initial values separated by commas or new lines.",
        "Up to 1,000 characters.",
        "Review the settings before running."
      ],
      "action": null
    },
    "mode": "explicit",
    "interactive": false
  },
  "execution": "Browser interface",
  "api": null,
  "browser_agent": {
    "supported": true,
    "name": "run_current_tool",
    "discovery": "WebMCP on the human page in a supporting browser",
    "verification": "See audit/API-AUDIT.md; availability is not verification"
  }
}

Page guide

List the terms of a recurrence exactly and, for linear recurrences with constant coefficients, find the closed form from the characteristic roots. You can also check your own guess against every computed term.

How to use Recurrence relation explorer
  1. Type the recurrence and initial values, such as a(n) = a(n-1) + a(n-2), a(0) = 0, a(1) = 1.
  2. Optionally add a guess, such as guess: a(n) = 2^n - 1, and choose how many terms to list.
  3. Run the explorer and read the terms, the characteristic roots and the closed form.
  4. Download the JSON report or the CSV of terms.

Useful next steps

FractionsCombinations and permutations calculatorPartition and composition calculator

Inputs are limited to 1,000 characters, order 10, forcing polynomials of degree 3 and 1 to 200 terms; exact values may grow to 1,024 digits. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of terms.

Closed forms

For a(n) = c1 a(n-1) + ... + ck a(n-k) + f(n) with a polynomial f, the characteristic polynomial is factored exactly. Each root r of multiplicity m contributes r^n, n r^n and so on, a polynomial particular solution handles f, and the constants come from the initial values. Rational roots and roots of the form p + q*sqrt(d) are supported. Method: Rosen, Discrete Mathematics and Its Applications, 8th edition, section 8.2 (2019).

Checks

The closed form is evaluated exactly at every computed term and must match. A guess is compared term by term; agreement on finitely many terms is evidence, not proof.

Other recurrences

Nonlinear recurrences such as a(n) = n*a(n-1) are expanded exactly but get no closed form. Complex characteristic roots and roots of cubic or higher factors are reported as not supported instead of guessed.