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pascal-fibonacci_triangle.pl
executable file
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pascal-fibonacci_triangle.pl
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#!/usr/bin/perl
# Daniel "Trizen" Șuteu
# Date: 25 March 2019
# https://github.com/trizen
# Generate the Pascal-Fibonacci triangle.
# Definition by Elliott Line, Mar 22 2019:
# Consider a version of Pascal's Triangle: a triangular array with a single 1 on row 0,
# with numbers below equal to the sum of the two numbers above it if and only if that sum
# appears in the Fibonacci sequence. If the sum is not a Fibonacci number, `1` is put in its place.
# OEIS sequence:
# https://oeis.org/A307069
use 5.010;
use strict;
use warnings;
use ntheory qw(is_square);
use experimental qw(signatures);
sub is_fibonacci($n) {
my $m = 5 * $n * $n;
is_square($m - 4) or is_square($m + 4);
}
my @row = (1);
my $rows = 40;
foreach my $n (1 .. $rows) {
my @t = (
map {
my $t = $row[$_] + $row[$_ + 1];
is_fibonacci($t) ? $t : 1;
} 0 .. ($n - ($n % 2)) / 2 - 1
);
say "@row";
# The triangle is symmetric
# See also: https://photos.app.goo.gl/q3981kei8LJyvzgZ9
my @u = reverse(@t);
if ($n % 2 == 0) {
shift @u;
}
@row = (1, @t, @u, 1);
}
__END__
1
1 1
1 2 1
1 3 3 1
1 1 1 1 1
1 2 2 2 2 1
1 3 1 1 1 3 1
1 1 1 2 2 1 1 1
1 2 2 3 1 3 2 2 1
1 3 1 5 1 1 5 1 3 1
1 1 1 1 1 2 1 1 1 1 1
1 2 2 2 2 3 3 2 2 2 2 1
1 3 1 1 1 5 1 5 1 1 1 3 1
1 1 1 2 2 1 1 1 1 2 2 1 1 1
1 2 2 3 1 3 2 2 2 3 1 3 2 2 1
1 3 1 5 1 1 5 1 1 5 1 1 5 1 3 1
1 1 1 1 1 2 1 1 2 1 1 2 1 1 1 1 1
1 2 2 2 2 3 3 2 3 3 2 3 3 2 2 2 2 1
1 3 1 1 1 5 1 5 5 1 5 5 1 5 1 1 1 3 1
1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1
1 2 2 3 1 3 2 2 2 2 2 2 2 2 2 3 1 3 2 2 1
1 3 1 5 1 1 5 1 1 1 1 1 1 1 1 5 1 1 5 1 3 1
1 1 1 1 1 2 1 1 2 2 2 2 2 2 2 1 1 2 1 1 1 1 1
1 2 2 2 2 3 3 2 3 1 1 1 1 1 1 3 2 3 3 2 2 2 2 1
1 3 1 1 1 5 1 5 5 1 2 2 2 2 2 1 5 5 1 5 1 1 1 3 1
1 1 1 2 2 1 1 1 1 1 3 1 1 1 1 3 1 1 1 1 1 2 2 1 1 1
1 2 2 3 1 3 2 2 2 2 1 1 2 2 2 1 1 2 2 2 2 3 1 3 2 2 1
1 3 1 5 1 1 5 1 1 1 3 2 3 1 1 3 2 3 1 1 1 5 1 1 5 1 3 1
1 1 1 1 1 2 1 1 2 2 1 5 5 1 2 1 5 5 1 2 2 1 1 2 1 1 1 1 1
1 2 2 2 2 3 3 2 3 1 3 1 1 1 3 3 1 1 1 3 1 3 2 3 3 2 2 2 2 1
1 3 1 1 1 5 1 5 5 1 1 1 2 2 1 1 1 2 2 1 1 1 5 5 1 5 1 1 1 3 1
1 1 1 2 2 1 1 1 1 1 2 2 3 1 3 2 2 3 1 3 2 2 1 1 1 1 1 2 2 1 1 1
1 2 2 3 1 3 2 2 2 2 3 1 5 1 1 5 1 5 1 1 5 1 3 2 2 2 2 3 1 3 2 2 1
1 3 1 5 1 1 5 1 1 1 5 1 1 1 2 1 1 1 1 2 1 1 1 5 1 1 1 5 1 1 5 1 3 1
1 1 1 1 1 2 1 1 2 2 1 1 2 2 3 3 2 2 2 3 3 2 2 1 1 2 2 1 1 2 1 1 1 1 1
1 2 2 2 2 3 3 2 3 1 3 2 3 1 5 1 5 1 1 5 1 5 1 3 2 3 1 3 2 3 3 2 2 2 2 1
1 3 1 1 1 5 1 5 5 1 1 5 5 1 1 1 1 1 2 1 1 1 1 1 5 5 1 1 5 5 1 5 1 1 1 3 1
1 1 1 2 2 1 1 1 1 1 2 1 1 1 2 2 2 2 3 3 2 2 2 2 1 1 1 2 1 1 1 1 1 2 2 1 1 1
1 2 2 3 1 3 2 2 2 2 3 3 2 2 3 1 1 1 5 1 5 1 1 1 3 2 2 3 3 2 2 2 2 3 1 3 2 2 1
1 3 1 5 1 1 5 1 1 1 5 1 5 1 5 1 2 2 1 1 1 1 2 2 1 5 1 5 1 5 1 1 1 5 1 1 5 1 3 1