# Category:Fibonacci Numbers

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This category contains results about Fibonacci Numbers.

Definitions specific to this category can be found in Definitions/Fibonacci Numbers.

The **Fibonacci numbers** are a sequence $\sequence {F_n}$ of integers which is formally defined recursively as:

- $F_n = \begin {cases} 0 & : n = 0 \\ 1 & : n = 1 \\ F_{n - 1} + F_{n - 2} & : \text {otherwise} \end {cases}$

for all $n \in \Z_{\ge 0}$.

## Subcategories

This category has the following 24 subcategories, out of 24 total.

### C

- Cassini's Identity (5 P)
- Catalan's Identity (3 P)

### E

- Euler-Binet Formula (8 P)

### F

- Fibonacci Nim (3 P)
- Fibonacci Numbers/Examples (24 P)
- Fibonacci Strings (11 P)

### G

### R

- Rabbit Problem (2 P)

### V

- Vajda's Identity (3 P)

### Z

- Zeckendorf Representation (4 P)

## Pages in category "Fibonacci Numbers"

The following 96 pages are in this category, out of 96 total.

### C

### F

- Fibonacci Number 2n equals Fibonacci Number n by Lucas Number n
- Fibonacci Number 3n in terms of Fibonacci Number n and Lucas Number 2n
- Fibonacci Number as Sum of Binomial Coefficients
- Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less
- Fibonacci Number by One Minus Golden Mean plus Fibonacci Number of Index One Less
- Fibonacci Number by Power of 2
- Fibonacci Number equal to Sum of Sequence of Cubes
- Fibonacci Number greater than Golden Section to Power less Two
- Fibonacci Number in terms of Larger Fibonacci Numbers
- Fibonacci Number in terms of Smaller Fibonacci Numbers
- Fibonacci Number is not Product of Two Smaller Fibonacci Numbers
- Fibonacci Number less than Golden Section to Power less One
- Fibonacci Number n+1 Minus Golden Mean by Fibonacci Number n
- Fibonacci Number of Even Index by Golden Mean Modulo 1
- Fibonacci Number of Index 2n as Sum of Squares of Fibonacci Numbers
- Fibonacci Number of Index 3n as Sum of Cubes of Fibonacci Numbers
- Fibonacci Number of Odd Index by Golden Mean Modulo 1
- Fibonacci Number plus Arbitrary Function in terms of Fibonacci Numbers
- Fibonacci Number plus Binomial Coefficient in terms of Fibonacci Numbers
- Fibonacci Number plus Constant in terms of Fibonacci Numbers
- Fibonacci Number whose Index is Plus or Minus Prime p is Multiple of p
- Fibonacci Number with Negative Index
- Fibonacci Number with Prime Index 2n+1 is Congruent to 5^n Modulo p
- Fibonacci Number with Prime Index is not necessarily Prime
- Fibonacci Numbers which equal the Square of their Index
- Fibonacci Numbers which equal their Index
- Fibonacci Prime has Prime Index except for 3

### G

- GCD of Consecutive Integers of General Fibonacci Sequence
- GCD of Fibonacci Numbers
- GCD of Generators of General Fibonacci Sequence is Divisor of All Terms
- General Fibonacci Number in terms of Fibonacci Numbers
- General Fibonacci Sequence whose Terms are all Composite
- Generating Function for Fibonacci Numbers

### N

### P

### R

- Rabbit Problem
- Ratio of Consecutive Fibonacci Numbers
- Reciprocal as Sum of Fibonacci Numbers by Negative Powers of 10
- Recurrence Formula for Sum of Sequence of Fibonacci Numbers
- Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers
- Recursively Defined Sequence/Examples/Minimum over k of Maximum of 1 plus Function of k and 2 plus Function of n-k
- Relation between Square of Fibonacci Number and Square of Lucas Number
- Remainder of Fibonacci Number Divided by Fibonacci Number is Plus or Minus Fibonacci Number
- Residue of Fibonacci Number Modulo Fibonacci Number

### S

- Second Order Fibonacci Number in terms of Fibonacci Numbers
- Sequence of Fibonacci Numbers ending in Index
- Sequence of Powers of Golden Mean
- Sine of Multiple of Pi by 2 plus i by Natural Logarithm of Golden Mean
- Smallest Positive Integer with 5 Fibonacci Partitions
- Square Fibonacci Number
- Sum of Alternating Sign Reciprocals of Sequence of Pairs of Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared
- Sum of Entries in Lesser Diagonal of Pascal's Triangle equal Fibonacci Number
- Sum of Even Sequence of Products of Consecutive Fibonacci Numbers
- Sum of Non-Consecutive Fibonacci Numbers
- Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers
- Sum of Reciprocals of Sequence of Pairs of Even Index Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared
- Sum of Sequence of Even Index Fibonacci Numbers
- Sum of Sequence of Fibonacci Numbers
- Sum of Sequence of Odd Index Fibonacci Numbers
- Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient
- Sum of Sequence of Product of Lucas Numbers with Powers of 2
- Sum of Sequence of Products of Consecutive Fibonacci Numbers
- Sum of Sequence of Squares of Fibonacci Numbers
- Sum of Squares of Consecutive Fibonacci Numbers
- Sum over k of n Choose k by Fibonacci Number with index m+k
- Sum over k of n Choose k by Fibonacci t to the k by Fibonacci t-1 to the n-k by Fibonacci m+k
- Summation over k to n of Product of kth with n-kth Fibonacci Numbers