1Math::NumSeq::SpiroFiboUnsaecrciC(o3n)tributed Perl DocuMmaetnht:a:tNiuomnSeq::SpiroFibonacci(3)
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6 Math::NumSeq::SpiroFibonacci -- recurrence around a square spiral
7
9 use Math::NumSeq::SpiroFibonacci;
10 my $seq = Math::NumSeq::SpiroFibonacci->new (cbrt => 2);
11 my ($i, $value) = $seq->next;
12
14 This is the spiro-Fibonacci numbers by Neil Fernandez. The sequence is
15 a recurrence
16 SF[0] = 0
18 SF[1] = 1
19 SF[i] = SF[i-1] + SF[i-k]
20 where the offset k is the closest point on the on the preceding loop of
22 a square spiral. The initial values are
23 0, 1, 1, ..., 1, 2, 3, 4, ... 61, 69, 78, 88, 98, 108, ...
25 starting i=0
26 On the square spiral this is
28 98-88-78-69-61-54-48
30 | |
31 108 10--9--8--7--6 42
32 | | | |
33 11 1--1--1 5 36
34 | | | | |
35 12 1 0--1 4 31
36 | | | |
37 13 1--1--2--3 27
38 | |
39 14-15-16-18-21-24
40 Value 36 on the right is 31+5, being the immediately preceding 31 and
42 the value on the next inward loop closest to that new 36 position.
43 At the corners the same inner value is used three times, so for example
45 42=36+6, then 48=42+6 and 54=48+6, all using the corner "6". For the
46 innermost loop SF[2] through SF[7] the "0" at the origin is the inner
47 value, hence the run of seven 1s at the start.
48 Absolute Differences
50 Optional "recurrence_type => 'absdiff'" changes the recurrence formula
51 to an absolute difference
52 SF[i] = abs (SF[i-1] - SF[i-k])
54 With the default initial values SF[0]=0 and SF[1]=1 this behaves as an
56 XOR, always giving 0 or 1.
57 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...
59 The result plotted around the square spiral is similar to some of the
61 cellular automaton patterns which work on xor feedback.
62 *** * * ** * ** ** * * * * * * **
64 * * ***** * *** * * * * ******** *****
65 ** * * **** * ******** * ** *
66 * ** * * ** * ** * * *
67 *** * ** * ** * * **** *
68 * ****** * *** * * **** * * *
69 ** * * ***** * *** * * ** ***
70 *** ** ** * * **o** ** ** * * *
71 * * * * ** * * * * * * * ******
72 ** **** * * * ** ********* *
73 ** **** * **** *** * * * *
74 * ** * ** * * * ******* ** ** *
75 ** ** * ** ** * * * * * * * * *
76 ** ** *** * * ** * ****** **** *
77 * * * ***** *** *** * * **
78 Initial Values
80 Optional "initial_0" and "initial_1" can give different initial i=0 and
81 i=1 values. For example "initial_0=>1, initial_1=>0" gives
82 1, 0, 1, 2, 3, 4, 5, 6, 6, 6, 6, 7, 8, 9, 11, 14, 17, 20, ...
84
86 See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
87 classes.
88 "$seq = Math::NumSeq::SpiroFibonacci->new ()"
90 Create and return a new sequence object.
91 "($i, $value) = $seq->next()"
93 Return the next index and value in the sequence.
94 When $value exceeds the range of a Perl unsigned integer the return
96 is promoted to a "Math::BigInt" to keep full precision.
97
99 Math::NumSeq, Math::NumSeq::Fibonacci
100 Math::PlanePath::SquareSpiral
102
104 <http://user42.tuxfamily.org/math-numseq/index.html>
105
107 Copyright 2012, 2013, 2014, 2016, 2019 Kevin Ryde
108 Math-NumSeq is free software; you can redistribute it and/or modify it
110 under the terms of the GNU General Public License as published by the
111 Free Software Foundation; either version 3, or (at your option) any
112 later version.
113 Math-NumSeq is distributed in the hope that it will be useful, but
115 WITHOUT ANY WARRANTY; without even the implied warranty of
116 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
117 General Public License for more details.
118 You should have received a copy of the GNU General Public License along
120 with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.
121perl v5.34.0 2021-07-22 Math::NumSeq::SpiroFibonacci(3)