Find the number of triangles with integer side lengths and perimeter p.


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Question Number 188301 by mr W last updated on 28/Feb/23

$F i n d t h e n u m b e r o f t r i a n g l e s w i t h$ $i n t e g e r s i d e l e n g t h s a n d p e r i m e t e r p .$
Commented by mr W last updated on 28/Feb/23

$a s Q 188239, b u t i n g e n e r a l c a s e .$
Commented by BaliramKumar last updated on 27/Feb/23

$w h y$
Commented by mr W last updated on 27/Feb/23

$f o r m o r e s o l u t i o n s!$
Commented by mr W last updated on 27/Feb/23

$i f y o u h a v e a s o l u t i o n, p l e a s e s h a r e .$
	Answered by mr W last updated on 04/Mar/23

	$t h i s i s m y s o l u t i o n :$ $s a y t h e s i d e s o f t h e t r i a n g l e a r e a, b, c$ $w i t h a ⩽ b ⩽ c, t h e n t h e q u e s t i o n i s t o$ $f i n d t h e n u m b e r o f s o l u t i o n s o f$ $f o l l o w i n g e q u a t i o n$ $a + b + c = p . . . (i)$ $w i t h 1 ⩽ a ⩽ b ⩽ c a n d a + b > c .$ $s a y$ $a = 1 + u w i t h u ⩾ 0$ $b = a + v = 1 + u + v w i t h v ⩾ 0$ $c = b + w = 1 + u + v + w w i t h w ⩾ 0$ $a n d$ $(1 + u) + (1 + u + v) > (1 + u + v + w)$ $\Rightarrow u > w - 1 \Rightarrow u ⩾ w$ $s a y u = w + s w i t h s ⩾ 0$ $t h e n t h e e q u a t i o n (i) b e c o m e s t o$ $(1 + w + s) + (1 + w + s + v) + (1 + w + s + v + w) = p$ $\Rightarrow 3 + 4 w + 3 s + 2 v = p . . . (I)$ $w i t h w, s, v ⩾ 0$ $t h e n u m b e r o f s o l u t i o n s o f e q n . (I)$ $i s t h e c o e f f i c i e n t o f x^{p} t e r m i n$ $t h e e x p a n s i o n o f$ $x^{3} (1 + x^{4} + x^{8} + . . .) (1 + x^{3} + x^{6} + . . .) (1 + x^{2} + x^{4} + . . .)$ $i . e . \frac{x^{3}}{(1 - x^{4}) (1 - x^{3}) (1 - x^{2})} .$ $i t s e x p a n s i o n s e e b e l o w .$ $e x a m p l e s :$ $p = 16 : t h e c o e f . o f x^{16} i s 5 .$ $\Rightarrow 5 t r i a n g l e s h a v e p = 16 .$ $p = 50 : t h e c o e f . o f x^{50} i s 52 .$ $\Rightarrow 52 t r i a n g l e s h a v e p = 50 .$ $p = 75 : t h e c o e f . o f x^{75} i s 127 .$ $\Rightarrow 127 t r i a n g l e s h a v e p = 75 .$
	Commented by mr W last updated on 04/Mar/23

	Commented by manxsol last updated on 27/Feb/23

	$B r a v o, S i r W . G r e a t w o r k .$ $t h a n k s f o r t h e e x p l a n a t i o n$
	Commented by mr W last updated on 28/Feb/23

	$t h i s i s a q u e s t i o n i n t h e n u m b e r$ $t h e o r y a b o u t t h e p a r t i t i o n o f a n$ $i n t e g e r .$
	Commented by mr W last updated on 28/Feb/23

	Commented by BaliramKumar last updated on 28/Feb/23

	$G r e a t S i r$
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