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.IX Title "Math::PlanePath::WythoffPreliminaryTriangle 3pm"
.TH Math::PlanePath::WythoffPreliminaryTriangle 3pm "2021-01-23" "perl v5.32.0" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
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.SH "NAME"
Math::PlanePath::WythoffPreliminaryTriangle \-\- Wythoff row containing X,Y recurrence
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 3
\& use Math::PlanePath::WythoffPreliminaryTriangle;
\& my $path = Math::PlanePath::WythoffPreliminaryTriangle\->new;
\& my ($x, $y) = $path\->n_to_xy (123);
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
This path is the Wythoff preliminary triangle by Clark
Kimberling,
.IX Xref "Kimberling, Clark"
.PP
.Vb 10
\&     13  | 105 118 131 144  60  65  70  75  80  85  90  95 100
\&     12  |  97 110  47  52  57  62  67  72  77  82  87  92    
\&     11  |  34  39  44  49  54  59  64  69  74  79  84        
\&     10  |  31  36  41  46  51  56  61  66  71  76            
\&      9  |  28  33  38  43  48  53  58  63  26                
\&      8  |  25  30  35  40  45  50  55  23                    
\&      7  |  22  27  32  37  42  18  20                        
\&      6  |  19  24  29  13  15  17                            
\&      5  |  16  21  10  12  14                                
\&      4  |   5   7   9  11                                    
\&      3  |   4   6   8                                        
\&      2  |   3   2                                            
\&      1  |   1                                                
\&    Y=0  |                                                    
\&         +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-
\&           X=0   1   2   3   4   5   6   7   8   9  10  11  12
.Ve
.PP
A given N is at an X,Y position in the triangle according to where row
number N of the Wythoff array \*(L"precurses\*(R" back to.  Each Wythoff row is a
Fibonacci recurrence.  Starting from the pair of values in the first and
second columns of row N it can be run in reverse by
.PP
.Vb 1
\&    F[i\-1] = F[i+i] \- F[i]
.Ve
.PP
It can be shown that such a reverse always reaches a pair Y and X with
Y>=1 and 0<=X<Y, hence making the triangular X,Y arrangement
above.
.PP
.Vb 8
\&    N=7 WythoffArray row 7 is 17,28,45,73,...
\&    go backwards from 17,28 by subtraction
\&       11 = 28 \- 17
\&        6 = 17 \- 11
\&        5 = 11 \- 6
\&        1 = 6 \- 5
\&        4 = 5 \- 1
\&    stop on reaching 4,1 which is Y=4,X=1 with Y>=1 and 0<=X<Y
.Ve
.PP
Conversely a coordinate pair X,Y is reckoned as the start of a Fibonacci
style recurrence,
.PP
.Vb 1
\&    F[i+i] = F[i] + F[i\-1]   starting F[1]=Y, F[2]=X
.Ve
.PP
Iterating these values gives a row of the Wythoff array
(Math::PlanePath::WythoffArray) after some initial iterations.  The N
value at X,Y is the row number of the Wythoff array which is reached.  Rows
are numbered starting from 1.  For example,
.PP
.Vb 3
\&    Y=4,X=1 sequence:       4, 1, 5, 6, 11, 17, 28, 45, ...
\&    row 7 of WythoffArray:                  17, 28, 45, ...
\&    so N=7 at Y=4,X=1
.Ve
.SH "FUNCTIONS"
.IX Header "FUNCTIONS"
See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for the behaviour common to all path
classes.
.ie n .IP """$path = Math::PlanePath::WythoffPreliminaryTriangle\->new ()""" 4
.el .IP "\f(CW$path = Math::PlanePath::WythoffPreliminaryTriangle\->new ()\fR" 4
.IX Item "$path = Math::PlanePath::WythoffPreliminaryTriangle->new ()"
Create and return a new path object.
.SH "OEIS"
.IX Header "OEIS"
Entries in Sloane's Online Encyclopedia of Integer Sequences related to
this path include
.Sp
.RS 4
<http://oeis.org/A165360> (etc)
.RE
.PP
.Vb 4
\&    A165360     X
\&    A165359     Y
\&    A166309     N by rows
\&    A173027     N on Y axis
.Ve
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Math::PlanePath,
Math::PlanePath::WythoffArray
.SH "HOME PAGE"
.IX Header "HOME PAGE"
<http://user42.tuxfamily.org/math\-planepath/index.html>
.SH "LICENSE"
.IX Header "LICENSE"
Copyright 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
.PP
This file is part of Math-PlanePath.
.PP
Math-PlanePath is free software; you can redistribute it and/or modify it
under the terms of the \s-1GNU\s0 General Public License as published by the Free
Software Foundation; either version 3, or (at your option) any later
version.
.PP
Math-PlanePath is distributed in the hope that it will be useful, but
\&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0
or \s-1FITNESS FOR A PARTICULAR PURPOSE.\s0  See the \s-1GNU\s0 General Public License for
more details.
.PP
You should have received a copy of the \s-1GNU\s0 General Public License along with
Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.