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Math-PlanePath-129
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::WythoffLines

NAME

Math::PlanePath::WythoffLines -- table of Fibonacci recurrences

SYNOPSIS

use Math::PlanePath::WythoffLines;
my $path = Math::PlanePath::WythoffLines->new;
my ($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 coordinate pair Y and X are the start of a Fibonacci style recurrence,

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

Any such sequence eventually becomes a row of the Wythoff array (Math::PlanePath::WythoffArray) after some number of initial iterations. The N value at X,Y is the row number of the Wythoff array containing sequence beginning Y and X. Rows are numbered starting from 1. Eg.

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

Conversely a given N is positioned in the triangle according to where row number N of the Wythoff array "precurses" by running the recurrence in reverse,

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

It can be shown that such a precurse 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 satisfying Y>=1 and 0<=X<Y

Phi Slope Blocks

The effect of each step backwards is to move to successive blocks of values, with slope golden ratio phi=(sqrt(5)+1)/2.

Suppose no backwards steps were applied, so Y,X were the first two values of Wythoff row N. In the example above that would be N=7 at Y=17,X=28. The first two values of the Wythoff array are

Y = W[0,r] = r-1 + floor(r*phi)       # r = row numbered from 1
X = W[1,r] = r-1 + 2*floor(r*phi)

So this would put N values on a line of slope Y/X = 1/phi = 0.618. The portion of that line which falls within 0<=X<Y

FUNCTIONS

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

$path = Math::PlanePath::WythoffLines->new ()

Create and return a new path object.

OEIS

The Wythoff array is in Sloane's Online Encyclopedia of Integer Sequences in various forms,

A165360     X
A165359     Y
A166309     N by rows

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/>.