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/>.

perl v5.32.0                                2021-Math::PlanePath::WythoffPreliminaryTriangle(3pm)