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# 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=Y, F=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

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

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