Gfis: Perl program and parameterization

2024-04-06T09:20:46Z

Perl program and parameterization

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==Triangles with integer sides, areas and in-radii==
* Sides with perimeter <code>p = a <= b <= c</code>: a = [https://oeis.org/A070080 A070080], b = [https://oeis.org/A070081 A070081], c = [https://oeis.org/A070082 A070082]
===[https://oeis.org/A070080/a070080.txt List] of [https://oeis.org/wiki/User:Reinhard_Zumkeller Reinhard Zumkeller] ===
(Reinhard was my colleague in April 1976 at the University of Dortmund and at Softlab GmbH in München.)
* [https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a070/A070080.java Java program] (for the jOEIS environment) that reproduces the list, with the area and in-radius values in addition.
* Perl program with identical output:
<pre>
#!perl
# @(#) $Id$
# Generate integer-sided triangles
# 2024-04-05, Georg Fischer
#
#:# Usage:
#:# perl A070080_gen.pl [start [[-]end]]
#:# start first perimeter (default 1)
#:# +end number of perimeters to be printed (default 1, or 55 if start=1)
#:# -end print from perimeter start to abs(end)
#--------
use strict;
# do not use integer! b.o. area and inradius
use warnings;

my $periStart = 1;
my $periEnd = 55;
my $debug = 0;
while (scalar(@ARGV) > 0 and ($ARGV[0] =~ m{\A[\-\+]})) {
my $opt = shift(@ARGV);
if (0) {
} elsif ($opt =~ m{d}) {
$debug = shift(@ARGV);
}
} # while
if (scalar(@ARGV) > 0) {
$periStart = shift(@ARGV);
}
if (scalar(@ARGV) > 0) {
$periEnd = shift(@ARGV);
} else {
if ($periStart > 1) {
$periEnd = $periStart;
}
}

my ($peri, $a, $b, $c, $r) =
(1, 1, 0, 2, 0);
$peri = $periStart;
$a = 1;
$b = 0;
if ($periStart > 1) {
&printSeparator();
}
my $n = 0;
my $oldPeri = $peri;
while ($peri <= $periEnd) {
$n ++;
&advance();
if ($peri != $oldPeri) {
&printSeparator();
$oldPeri = $peri;
}
if ($peri <= $periEnd) {
print sprintf("| %5d | %5d |%4d%4d%4d |", $n, $peri, $a, $b, $c);
print sprintf("%6d |%6d |", &gcd($a, &gcd($b, $c)), $a**2 + $b**2 - $c**2);
my $s = $peri/2;
my $H = sprintf("%12.6f", sqrt($s*($s - $a)*($s - $b)*($s - $c)));
my $I = sprintf( "%8.6f", sqrt( ($s - $a)*($s - $b)*($s - $c)/$s));
print " s" if $a < $b && $b < $c;
print " i" if $a == $b || $b == $c;
if (0) {
} elsif (&isPrime($a) && &isPrime($b) && &isPrime($c)) {
print " p";
} elsif (&gcd($a, &gcd($b, $c)) == 1) {
print " r";
} else {
print " ";
}
print " A" if $a**2 + $b**2 > $c**2;
print " R" if $a**2 + $b**2 == $c**2;
print " O" if $a**2 + $b**2 < $c**2;
print "" . (($H =~ m{\.000000}) ? " H" : " ");
print "" . (($I =~ m{\.000000}) ? " I" : " ");
print " | $H $I";
print "\n";
}
} # for $n

#----
sub printSeparator {
print "+-------+-------+-------------+-------+-------+-----------+\n";
}

sub advance {
my $busy = 1;
while ($busy) {
$b ++;
$c = $peri - $a - $b;
print sprintf("%3d: %3d %3d %3d P\n", $peri, $a, $b, $c) if $debug > 0;
if ($a <= $b and $b <= $c && $a + $b > $c) {
$busy = 0;
} elsif ($b > $c) {
$a ++;
$r = $peri - $a;
$b = $a;
$c = $peri - $a - $b;
print sprintf("%3d: %3d %3d %3d Q\n", $peri, $a, $b, $c) if $debug > 0;
if ($a <= $b and $b <= $c && $a + $b > $c) {
$busy = 0;
} elsif ($b + $c <= $a) {
$peri ++;
$a = 1;
$r = $peri - $a;
$b = $a;
$c = $peri - $a - $b;
print sprintf("%3d: %3d %3d %3d R\n", $peri, $a, $b, $c) if $debug > 0;
if ($a <= $b and $b <= $c && $a + $b > $c) {
$busy = 0;
}
}
}
} # while
} # advance

# from https://gist.github.com/EricCrosson/8b5dc6a15679f159f7fc
sub gcd {
my ($m, $n) = @_;
if ($n == 0) {
return $m;
}
return gcd($n, $m % $n);
}

# https://stackoverflow.com/questions/32675465/perl-finding-out-if-a-given-number-is-a-prime-number
sub isPrime {
my ($num) = @_;
# 1 1 1 1 1 1 1 1 1 1 2
# 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
my @pc = (0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1);
return $pc[$num] if $num < scalar(@pc);
return 0 if $num % 2 == 0 ;
for (my $i = 3; $i <= $num/2; $i += 2) {
return 0 if ($num % $i == 0)
}
return 1;
}

__DATA__
s = scalene: a < b < c |
| i = isosceles: a = b or b = c |
| |
| r = a, b, and c are relatively prime |
| p = a, b, and c are primes (p < r) |
| |
| A = acute: a^2 + b^2 > c^2 |
| R = right: a^2 + b^2 = c^2 |
| O = obtuse: a^2 + b^2 < c^2 |
+-----+-------+----------+-------+-------+-----------------+
|Sequ-|A070083|A070080-82|A070084|A070085| |
| Nr. | a+b+c | a b c | gcd | | properties |
+-----+-------+----------+-------+-------+-----------------+
| 1 | 3 | 1 1 1 | 1 | 1 | i r A |
|- - - - - - - - - - - - - - - - - - - -|
| 2 | 5 | 1 2 2 | 1 | 1 | i r A |
|- - - - - - - - - - - - - - - - - - - -|
| 3 | 6 | 2 2 2 | 2 | 4 | i p A |
|
</pre>
===Parameterization of related sequences ===
<pre>
A070080 SIDE_A s -> isTriangle(s) Smallest side of integer triangles [a(n) <= A070081(n) <= A070082(n)], sorted by perimeter, lexicographically ordered.
A070081 SIDE_B s -> isTriangle(s) Middle side of integer triangles [A070080(n) <= a(n) <= A070082(n)], sorted by perimeter, sides lexicographically ordered.
A070082 SIDE_C s -> isTriangle(s) Largest side of integer triangles [A070080(n) <= A070081(n) <= a(n)], sorted by perimeter, sides lexicographically ordered.
A070083 PERIMETER s -> isTriangle(s) Perimeters of integer triangles, sorted by perimeter, sides lexicographically ordered.
A070084 GCD s -> isTriangle(s) Greatest common divisor of sides of integer triangles [A070080(n), A070081(n), A070082(n)], sorted by perimeter, sides lexicographically ordered.
A070085 SHAPE s -> isTriangle(s) a(n) = A070080(n)^2 + A070081(n)^2 - A070082(n)^2.
A070086 AREA s -> isTriangle(s) Areas of integer triangles [A070080(n), A070081(n), A070082(n)], rounded values.
A070088 COUNT s -> hasPrimeSides(s) Number of integer-sided triangles with perimeter n and prime sides.
A070090 COUNT s -> isScalene(s) && hasPrimeSides(s)
A070091 COUNT s -> isIsosceles(s) && hasCoPrimeSides(s)
A070092 COUNT s -> isIsosceles(s) && hasPrimeSides(s)
A070093 COUNT s -> isAcute(s)
A070094 COUNT s -> isAcute(s) && hasCoPrimeSides(s)
A070095 COUNT s -> isAcute(s) && hasPrimeSides(s)
A070096 COUNT s -> isAcute(s) && isScalene(s) && hasCoPrimeSides(s)
A070097 COUNT s -> isAcute(s) && isScalene(s) && hasPrimeSides(s)
A070098 COUNT s -> isAcute(s) && isIsosceles(s)
A070099 COUNT s -> isAcute(s) && isIsosceles(s) && hasCoPrimeSides(s)
A070100 COUNT s -> isAcute(s) && isIsosceles(s) && hasPrimeSides(s)
A070101 COUNT s -> isObtuse(s)
A070102 COUNT s -> isObtuse(s) && hasCoPrimeSides(s)
A070103 COUNT s -> isObtuse(s) && hasPrimeSides(s)
A070104 COUNT s -> isObtuse(s) && isScalene(s) && hasCoPrimeSides(s)
A070105 COUNT s -> isObtuse(s) && isScalene(s) && hasPrimeSides(s)
A070106 COUNT s -> isObtuse(s) && isIsosceles(s)
A070107 COUNT s -> isObtuse(s) && isIsosceles(s) && hasCoPrimeSides(s)
A070108 COUNT s -> isObtuse(s) && isIsosceles(s) && hasPrimeSides(s)
A070109 COUNT s -> isRight(s) && hasCoPrimeSides(s)
A070110 INDEX s -> hasCoPrimeSides(s)
A070111 INDEX s -> hasPrimeSides(s)
A070112 INDEX s -> isScalene(s)
A070113 INDEX s -> isScalene(s) && hasCoPrimeSides(s)
A070114 INDEX s -> isScalene(s) && hasPrimeSides(s)
A070115 INDEX s -> isIsosceles(s)
A070116 INDEX s -> isIsosceles(s) && hasCoPrimeSides(s)
A070117 INDEX s -> isIsosceles(s) && hasPrimeSides(s)
A070118 INDEX s -> isAcute(s)
A070119 INDEX s -> isAcute(s) && hasCoPrimeSides(s)
A070120 INDEX s -> isAcute(s) && hasPrimeSides(s)
A070121 INDEX s -> isAcute(s) && isScalene(s)
A070122 INDEX s -> isAcute(s) && isScalene(s) && hasCoPrimeSides(s)
A070123 INDEX s -> isAcute(s) && isScalene(s) && hasPrimeSides(s)
A070124 INDEX s -> isAcute(s) && isIsosceles(s)
A070125 INDEX s -> isAcute(s) && isIsosceles(s) && hasCoPrimeSides(s)
A070126 INDEX s -> isAcute(s) && isIsosceles(s) && hasPrimeSides(s)
A070127 INDEX s -> isObtuse(s)
A070128 INDEX s -> isObtuse(s) && hasCoPrimeSides(s)
A070129 INDEX s -> isObtuse(s) && hasPrimeSides(s)
A070130 INDEX s -> isObtuse(s) && isScalene(s)
A070131 INDEX s -> isObtuse(s) && isScalene(s) && hasCoPrimeSides(s)
A070132 INDEX s -> isObtuse(s) && isScalene(s) && hasPrimeSides(s)
A070133 INDEX s -> isObtuse(s) && isIsosceles(s)
A070134 INDEX s -> isObtuse(s) && isIsosceles(s) && hasCoPrimeSides(s)
A070135 INDEX s -> isObtuse(s) && isIsosceles(s) && hasPrimeSides(s)
A070136 INDEX s -> isRight(s)
A070137 INDEX s -> isRight(s) && hasCoPrimeSides(s)
A070138 COUNT s -> hasCoPrimeSides(s) && hasIntArea(s)
A070139 COUNT s -> isIsosceles(s) && hasIntArea(s)
A070140 COUNT s -> isAcute(s) && hasIntArea(s)
A070141 COUNT s -> isObtuse(s) && hasIntArea(s)
A070142 INDEX s -> hasIntArea(s)
A070143 INDEX s -> hasCoPrimeSides(s) && hasIntArea(s)
A070144 INDEX s -> isScalene(s) && hasIntArea(s)
A070145 INDEX s -> isIsosceles(s) && hasIntArea(s)
A070146 INDEX s -> isAcute(s) && hasIntArea(s)
A070147 INDEX s -> isObtuse(s) && hasIntArea(s)
A070148 INDEX s -> hasTrigonalArea(s)
A070149 AREA s -> hasIntArea(s)
A070200 INRAD s -> isTriangle(s)
A070201 COUNT s -> hasIntInRadius(s)
A070202 COUNT s -> !hasCoPrimeSides(s) && hasIntInRadius(s)
A070203 COUNT s -> isScalene(s) && hasIntInRadius(s)
A070204 COUNT s -> isIsosceles(s) && hasIntInRadius(s)
A070205 COUNT s -> isAcute(s) && hasIntInRadius(s)
A070206 COUNT s -> isObtuse(s) && hasIntInRadius(s)
A070208 COUNT s -> hasIntArea(s) && !hasIntInRadius(s)
A070209 INDEX s -> hasIntInRadius(s)
A070210 INRAD s -> hasIntInRadius(s)

A051493 COUNT s -> hasCoPrimeSides(s)
A051516 COUNT s -> hasIntArea(s)
A059169 COUNT s -> isIsosceles(s)
A070150 filter 1 new A070149() TRIANGULAR Triangular areas of integer Heronian triangles.
A070151 tuptraf 1 (n, s) -> s[0].*(s[1])./(2) "" new A002330().skip(1), new A002331().skip(1)
A135622 parmof3 1 A070080 SQUARE16 s -> isTriangle(s) 16*Area^2 of integer triangles.
</pre>

OEIS/A070080 - Revision history

Gfis: Perl program and parameterization