A024153 Number of integer-sided triangles with sides a,b,c, a

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 1, 0, 1, 0, 2, 0, 3, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 0, 8, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 4, 0, 3, 0, 1

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

No such triangles with odd perimeter (see A051516).

Records occur at: 1, 12, 36, 54, 84, 108, 192, 216, 294, 324, 378, 420, 432, 540, 588, 756, 972, 1176, 1452, 1764, 1944, 2028, 2352, 2904, 2916, 3024, 3072, 3402, 3468, 3780, 3888, 4116, 5292, 6348, 6804, 8748, 10164, ... - Antti Karttunen, Sep 25 2018

Links

• Antti Karttunen, Table of n, a(n) for n = 1..12021 (terms 1..1000 from Seiichi Manyama)
• Eric Weisstein's World of Mathematics, Heronian Triangle.

Crossrefs

Cf. A051516, A070139, A070203, A188158, A301384.

Programs

PARI

A024153(n) = if(n%2,0,my(k=0, t, p=n/2); for(a=1,n,for(b=1+max(a,(p-a)),n-(a+1),my(c=n-(a+b)); if((c<=b),break); if(((t = (p*(p-a)*(p-b)*(p-c))) > 0)&&issquare(t),k++))); (k)); \\ Antti Karttunen, Sep 25 2018

Extensions

a(100) corrected by Seiichi Manyama, Sep 13 2018