# 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.

# 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 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/>.