pad

Sequence/Iteration Support

This module provides powerful sequence generation capabilities for the PAD framework, supporting mathematical sequences, number theory sequences, random generation, and array operations.

Overview

The sequence module (internally prefixed with pq) enables templates to generate and manipulate numeric sequences like Fibonacci, primes, ranges, and many specialized mathematical sequences. It provides a comprehensive system for building sequences with customizable parameters (from/to, increment, limits), applying transformations (sort, reverse, dedup), and extracting results (sum, average, count).

Directory Structure

Core Components

• inits/ - Initialization and parameter handling
  • Parse sequence parameters (from, to, increment, rows, etc.)
  • Find and validate sequence definitions
  • Set up loop limits and constraints
• build/ - Sequence construction
  • given.php - Process named sequences
  • parm.php - Parameter extraction
  • store.php - Store built sequences
  • vars.php - Variable management
  • randomly/ - Random sequence building
• exits/ - Result extraction and output
  • data.php - Return sequence data
  • return.php - Main return logic
  • store/ - Store results to variables
  • extra/ - Additional processing (chain, actions, etc.)
• plays/ - Sequence execution
  • parm.php - Play parameters
  • play/ - Different play modes (bool, function, order)

Sequence Types (types/)

Over 70 mathematical and number theory sequences including:

Arithmetic Sequences

• add, multiply, divide, subtract, modulo - Basic operations
• exponentiation, power - Exponential sequences
• repeat, step - Simple patterns

Number Properties

• even, odd - Parity-based sequences
• prime, antiprime, emirp, semiprime - Prime-related
• composite - Composite numbers
• square, cubic, biquadratic - Power sequences
• palindrome - Palindromic numbers
• happy, lucky - Special number properties

Famous Sequences

• fibonacci, lucas, tribonacci - Fibonacci family
• catalan - Catalan numbers
• bell - Bell numbers
• pell - Pell numbers
• perrin - Perrin sequence
• golomb, kolakoski - Self-describing sequences

Polygonal Numbers

• triangular, square, pentagonal, hexagonal, heptagonal, octagonal
• decagonal, heptadecagonal, enneadecagonal, icosihenagonal
• gnomonic, pronic

3D Figurate Numbers

• tetrahedral, octahedral, caterer

Other Sequences

• recaman, sylvester, newman-conway - Recursive sequences
• mersenne, cullen, kynea - Named after mathematicians
• perfect, powerful, polite, harshad, kaprekar - Special properties
• oeis - Reference OEIS sequences

Utilities

• range, list, loop - Basic iteration
• random, chance - Random number generation
• identity, eval - Special functions

Logical Operations

• and, or, xor, nand, nor, xnor, not, negation - Boolean logic
• ceil, floor, round - Rounding operations

Actions (actions/)

Transform and analyze sequences:

Aggregation

• sum.php - Sum all values
• average.php - Calculate mean
• product.php - Multiply all values
• count.php - Count elements
• median.php - Find median

Selection

• first.php - Get first element
• last.php - Get last element
• trim.php - Remove elements
• splice.php - Extract/insert elements

Transformation

• sort.php - Sort ascending
• reverse.php - Reverse order
• shuffle.php, randomize.php - Random reordering
• shift.php - Rotate elements
• dedup.php, distinct.php - Remove duplicates

Combination

• append.php, prepend.php - Add elements
• combine.php - Merge sequences

Ordering

• order/keys.php, order/values.php - Order by keys or values
• negative/keys.php, negative/values.php - Negative ordering

Structure

• assoc.php - Create associative arrays
• set.php - Set operations

Options (options/types/)

Configure sequence generation:

Range Control

• from.php, to.php - Define range boundaries
• min.php, max.php - Value constraints
• increment.php - Step size

Limit Control

• rows.php - Maximum number of elements
• stop.php - Stop condition
• skip.php - Skip initial elements
• sole.php - Single value mode

Behavior

• atLeastOnce.php - Ensure minimum execution
• try.php - Attempt count
• operation.php - Operation mode
• single.php - Single element mode

Ordering

• orderly.php - Ordered processing
• negative.php - Reverse ordering
• left.php, right.php - Direction control

Duplicates

• unique.php - Remove duplicates
• duplicates.php - Handle duplicates
• keep.php - Keep specific elements
• remove.php - Remove specific elements

Actions & Storage

• action.php - Apply action
• build.php - Build mode
• make.php - Creation mode
• store.php - Store result
• toData.php - Output to data variable
• name.php - Named sequence
• pull.php, push.php - Stack operations

Special

• both.php - Both keys and values
• count.php - Count mode
• randomly.php - Random mode

Key Features

Flexible Sequence Generation

• Generate any mathematical sequence by type name
• Customize range (from/to), increment, and limits
• Support for both ordered and random generation
• Named sequences for reuse

Powerful Transformations

• Chain multiple actions (sort, reverse, dedup, etc.)
• Aggregate operations (sum, average, count)
• Array manipulation (append, prepend, combine)
• Extract subsets (first, last, splice)

Advanced Parameters

• from/to: Define range boundaries
• increment: Step size between values
• rows: Limit number of results
• stop: Maximum value constraint
• skip: Skip initial values
• try: Attempt limits for validation
• unique: Remove duplicate values
• negative: Reverse ordering
• randomly: Random selection mode

Storage and Reuse

• Store sequences in named variables (build)
• Save results to data store (toData)
• Pull/push stack operations
• Reference built sequences

Integration with PAD Framework

The sequence module extends PAD’s template capabilities:

1. Template Tags: Sequences accessible via template tags with parameters
2. Data Generation: Generate data arrays for iteration
3. Calculations: Perform mathematical operations in templates
4. Random Data: Generate random datasets for testing/demos
5. Number Theory: Access mathematical sequences for specialized applications

Usage Pattern

Sequences are invoked through PAD template tags:

{sequence type="fibonacci" from="1" to="10"}
{sequence type="prime" rows="20" action="sum"}
{sequence type="range" from="1" to="100" increment="5" action="average"}
{sequence type="random" from="1" to="100" rows="10" unique="true"}

Parameters control generation, actions transform results, and outputs can be stored for later use.

Example Workflows

1. Generate Fibonacci sequence: Type=fibonacci, from=1, to=10
2. Sum first 20 primes: Type=prime, rows=20, action=sum
3. Random unique numbers: Type=random, from=1, to=100, rows=10, unique=true
4. Even numbers descending: Type=even, from=2, to=20, negative=true
5. Average of range: Type=range, from=1, to=100, action=average

This module demonstrates PAD’s extensibility, providing a complete mathematical sequence engine within the template system.

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