OEIS/A070080
Triangles with integer sides, areas and in-radii
List of Reinhard Zumkeller
(Reinhard was my colleague in April 1976 at the University of Dortmund and at Softlab GmbH in München.)
- Java program (for the jOEIS environment) that reproduces the list, with the area and in-radius values in addition.
- Perl program with identical output:
#!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 |
|
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.