Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sums of products of binomial coefficients mod 2 and run length transforms of sequences

Chai Wah Wu

TL;DR

This work analyzes sums of products of binomial coefficients modulo $2$ and shows these sums are exactly the run length transforms of classical sequences. By exploiting Lucas' theorem and a bitwise/parity framework with $F(n,k)$ and $g(n,k)$, it derives recurrence structures that reduce complex parity sums to compact index-based relations. The main results identify run length transforms for the Fibonacci sequence, the positive integers, extended Lucas numbers, and Narayana's cows sequence, and extend the framework to third- and fourth-order recurrences, including fixed points. The results connect the parity structure of Pascal's triangle (Sierpiński triangle) to run length transforms and cellular automata, offering compact representations with potential applications in combinatorics and number theory.

Abstract

We study properties of functions of binomial coefficients mod 2 and derive a set of recurrence relations for sums of products of binomial coefficients mod 2. We show that they result in sequences that are the run length transforms of well known basic sequences. In particular, we obtain formulas for the run length transform of the positive integers, Fibonacci numbers, extended Lucas numbers and Narayana's cows sequence.

Sums of products of binomial coefficients mod 2 and run length transforms of sequences

TL;DR

This work analyzes sums of products of binomial coefficients modulo and shows these sums are exactly the run length transforms of classical sequences. By exploiting Lucas' theorem and a bitwise/parity framework with and , it derives recurrence structures that reduce complex parity sums to compact index-based relations. The main results identify run length transforms for the Fibonacci sequence, the positive integers, extended Lucas numbers, and Narayana's cows sequence, and extend the framework to third- and fourth-order recurrences, including fixed points. The results connect the parity structure of Pascal's triangle (Sierpiński triangle) to run length transforms and cellular automata, offering compact representations with potential applications in combinatorics and number theory.

Abstract

We study properties of functions of binomial coefficients mod 2 and derive a set of recurrence relations for sums of products of binomial coefficients mod 2. We show that they result in sequences that are the run length transforms of well known basic sequences. In particular, we obtain formulas for the run length transform of the positive integers, Fibonacci numbers, extended Lucas numbers and Narayana's cows sequence.

Paper Structure

This paper contains 16 sections, 30 theorems, 17 equations, 1 table.

Table of Contents

  1. Introduction
  2. Run length transform
  3. Recurrence relations of products of binomial coefficients mod $2$
  4. Sums of products of binomial coefficients mod $2$
  5. Run length transform of the Fibonacci sequence
  6. Run length transform of the truncated Fibonacci sequence
  7. Run length transform of $\{1,1,2,4,8,16,32,\dots\}$
  8. Run length transform of $\{1,2,2,2,2,2,\dots\}$
  9. Run length transform of the positive integers
  10. A fixed point of the run length transform
  11. Third order recurrences
  12. Run length transform of Narayana's cows sequence
  13. Run length transform of $1,1, 2,2, 3,3,4,4,5,5,\dots$
  14. Fourth order recurrences
  15. Run length transform of $1,1,2,1,3,4,7,11,18,\dots$
  16. ...and 1 more sections

Key Result

Theorem 1

$\left(\right) \equiv 0 \mod 2$ if and only if $k \wedge ( \neg n)\neq 0$.

Theorems & Definitions (58)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Definition 1
  • Theorem 4
  • Definition 2
  • Theorem 5
  • proof
  • Lemma 2
  • ...and 48 more