1Math::PlanePath::WythofUfsPerrelCiomnitnraiMrbayutTthre:id:aPnPlgealrnele(P3Da)otchu:m:eWnyttahtoifofnPreliminaryTriangle(3)
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NAME

6       Math::PlanePath::WythoffPreliminaryTriangle -- Wythoff row containing
7       X,Y recurrence
8

SYNOPSIS

10        use Math::PlanePath::WythoffPreliminaryTriangle;
11        my $path = Math::PlanePath::WythoffPreliminaryTriangle->new;
12        my ($x, $y) = $path->n_to_xy (123);
13

DESCRIPTION

15       This path is the Wythoff preliminary triangle by Clark Kimberling,
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17            13  | 105 118 131 144  60  65  70  75  80  85  90  95 100
18            12  |  97 110  47  52  57  62  67  72  77  82  87  92
19            11  |  34  39  44  49  54  59  64  69  74  79  84
20            10  |  31  36  41  46  51  56  61  66  71  76
21             9  |  28  33  38  43  48  53  58  63  26
22             8  |  25  30  35  40  45  50  55  23
23             7  |  22  27  32  37  42  18  20
24             6  |  19  24  29  13  15  17
25             5  |  16  21  10  12  14
26             4  |   5   7   9  11
27             3  |   4   6   8
28             2  |   3   2
29             1  |   1
30           Y=0  |
31                +-----------------------------------------------------
32                  X=0   1   2   3   4   5   6   7   8   9  10  11  12
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34       A given N is at an X,Y position in the triangle according to where row
35       number N of the Wythoff array "precurses" back to.  Each Wythoff row is
36       a Fibonacci recurrence.  Starting from the pair of values in the first
37       and second columns of row N it can be run in reverse by
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39           F[i-1] = F[i+i] - F[i]
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41       It can be shown that such a reverse always reaches a pair Y and X with
42       Y>=1 and 0<=X<Y, hence making the triangular X,Y arrangement above.
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44           N=7 WythoffArray row 7 is 17,28,45,73,...
45           go backwards from 17,28 by subtraction
46              11 = 28 - 17
47               6 = 17 - 11
48               5 = 11 - 6
49               1 = 6 - 5
50               4 = 5 - 1
51           stop on reaching 4,1 which is Y=4,X=1 with Y>=1 and 0<=X<Y
52
53       Conversely a coordinate pair X,Y is reckoned as the start of a
54       Fibonacci style recurrence,
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56           F[i+i] = F[i] + F[i-1]   starting F[1]=Y, F[2]=X
57
58       Iterating these values gives a row of the Wythoff array
59       (Math::PlanePath::WythoffArray) after some initial iterations.  The N
60       value at X,Y is the row number of the Wythoff array which is reached.
61       Rows are numbered starting from 1.  For example,
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63           Y=4,X=1 sequence:       4, 1, 5, 6, 11, 17, 28, 45, ...
64           row 7 of WythoffArray:                  17, 28, 45, ...
65           so N=7 at Y=4,X=1
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FUNCTIONS

68       See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path
69       classes.
70
71       "$path = Math::PlanePath::WythoffPreliminaryTriangle->new ()"
72           Create and return a new path object.
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OEIS

75       Entries in Sloane's Online Encyclopedia of Integer Sequences related to
76       this path include
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78           <http://oeis.org/A165360> (etc)
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80           A165360     X
81           A165359     Y
82           A166309     N by rows
83           A173027     N on Y axis
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SEE ALSO

86       Math::PlanePath, Math::PlanePath::WythoffArray
87

HOME PAGE

89       <http://user42.tuxfamily.org/math-planepath/index.html>
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LICENSE

92       Copyright 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
93
94       This file is part of Math-PlanePath.
95
96       Math-PlanePath is free software; you can redistribute it and/or modify
97       it under the terms of the GNU General Public License as published by
98       the Free Software Foundation; either version 3, or (at your option) any
99       later version.
100
101       Math-PlanePath is distributed in the hope that it will be useful, but
102       WITHOUT ANY WARRANTY; without even the implied warranty of
103       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
104       General Public License for more details.
105
106       You should have received a copy of the GNU General Public License along
107       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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111perl v5.36.0                    Ma2t0h2:3:-P0l1a-n2e0Path::WythoffPreliminaryTriangle(3)
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