Math::PlanePath::WythoffPreliminaryTriangle -- Wythoff row containing X,Y recurrence
use Math::PlanePath::WythoffPreliminaryTriangle; my $path = Math::PlanePath::WythoffPreliminaryTriangle->new; my ($x, $y) = $path->n_to_xy (123);
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
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.
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
Math::PlanePath, Math::PlanePath::WythoffArray
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2013, 2014, 2015, 2016, 2017 Kevin Ryde
This file is part of Math-PlanePath.
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