/ 
Math-NumSeq-73
River stage zero No dependents
/ Math::NumSeq::SpiroFibonacci

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.

SEE ALSO

Math::NumSeq, Math::NumSeq::Fibonacci

Math::PlanePath::SquareSpiral

HOME PAGE

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

LICENSE

Copyright 2012, 2013, 2014, 2016 Kevin Ryde

Math-NumSeq is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-NumSeq is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.