Reference

Math::PlanePath::WythoffPreliminaryTriangle(3pm)

NAME

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

SYNOPSIS

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

DESCRIPTION

This path is the Wythoff preliminary triangle by Clark Kimberling,

     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

A given N is at an X,Y position in the triangle according to where row number N of the Wythoff array "precurses" 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

    F[i-1] = F[i+i] - F[i]

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.

    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

Conversely a coordinate pair X,Y is reckoned as the start of a Fibonacci style recurrence,

    F[i+i] = F[i] + F[i-1]   starting F[1]=Y, F[2]=X

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,

    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

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.

"$path = Math::PlanePath::WythoffPreliminaryTriangle->new ()"

Create and return a new path object.

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

<http://oeis.org/A165360> (etc)

    A165360     X
    A165359     Y
    A166309     N by rows
    A173027     N on Y axis

SEE ALSO

Math::PlanePath, Math::PlanePath::WythoffArray

HOME PAGE

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

LICENSE

Copyright 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde

This file is part of Math-PlanePath.

Math-PlanePath 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-PlanePath 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-PlanePath. If not, see <http://www.gnu.org/licenses/>.

History