Kevin Ryde > Math-NumSeq-64 > Math::NumSeq::SpiroFibonacci

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Module Version: 64   Source   Latest Release: Math-NumSeq-69

# NAME

Math::NumSeq::SpiroFibonacci -- recurrence around a square spiral

# SYNOPSIS

``` use Math::NumSeq::SpiroFibonacci;
my \$seq = Math::NumSeq::SpiroFibonacci->new (cbrt => 2);
my (\$i, \$value) = \$seq->next;```

# DESCRIPTION

This is the spiro-Fibonacci numbers by Neil Fernandez. The sequence is a recurrence

```    SF[0] = 0
SF[1] = 1
SF[i] = SF[i-1] + SF[i-k]```

where the offset k is the closest point on the on the preceding loop of a square spiral. The initial values are

```    0, 1, 1, ..., 1, 2, 3, 4, ... 61, 69, 78, 88, 98, 108, ...
starting i=0```

On the square spiral this is

```     98-88-78-69-61-54-48
|                 |
108 10--9--8--7--6 42
|  |           |  |
11  1--1--1  5 36
|  |     |  |  |
12  1  0--1  4 31
|  |        |  |
13  1--1--2--3 27
|              |
14-15-16-18-21-24```

Value 36 on the right is 31+5, being the immediately preceding 31 and the value on the next inward loop closest to that new 36 position.

At the corners the same inner value is used three times, so for example 42=36+6, then 48=42+6 and 54=48+6, all using the corner "6". For the innermost loop SF[2] through SF[7] the "0" at the origin is the inner value, hence the run of seven 1s at the start.

## Absolute Differences

Optional `recurrence_type => 'absdiff'` changes the recurrence formula to an absolute difference

`    SF[i] = abs (SF[i-1] - SF[i-k])`

With the default initial values SF[0]=0 and SF[1]=1 this behaves as an XOR, always giving 0 or 1.

`    0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...`

The result plotted around the square spiral is similar to some of the cellular automaton patterns which work on xor feedback.

```    *** *    *  **       *     **  **      * * * *  * * **
* *  *****   *        *** * * * *      ******** *****
**   * * ****         * ********      *       **    *
*    **  * *          **      *      **      * *   *
***     *   **           *     **      * *     ****  *
* ******    *            *** * *      ****    *   * *
** * * *****             * ***       *   *   **  ***
*** **  ** * *              **o**      **  **  * * *
* *  *   *  **               * * *     * * * * ******
**   ****   *              *  * **    *********     *
** **** * ****              ***    *   *        *    *
* ** * **  * *              * *******  **       **   *
** **  *   **              ** * * * * * *      * *  *
** ** ***    *              *  **  * ******     **** *
*  *  * *****              *** ***        *    *   **```

## Initial Values

Optional `initial_0` and `initial_1` can give different initial i=0 and i=1 values. For example `initial_0=>1, initial_1=>0` gives

`    1, 0, 1, 2, 3, 4, 5, 6, 6, 6, 6, 7, 8, 9, 11, 14, 17, 20, ...`

# FUNCTIONS

See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.

`\$seq = Math::NumSeq::SpiroFibonacci->new ()`

Create and return a new sequence object.

`(\$i, \$value) = \$seq->next()`

Return the next index and value in the sequence.

When `\$value` exceeds the range of a Perl unsigned integer the return is promoted to a `Math::BigInt` to keep full precision.

Math::PlanePath::SquareSpiral

http://user42.tuxfamily.org/math-numseq/index.html