A024153 Number of integer-sided triangles with sides a,b,c, a<b<c, a+b+c=n that have integer area.

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

1,36

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.

Prog

(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

Crossrefs

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

Sequence in context: A070203 A070201 A070138 * A354527 A341523 A079127

Adjacent sequences: A024150 A024151 A024152 * A024154 A024155 A024156

Keyword

nonn

Author

Clark Kimberling

Extensions

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

Status

approved