A070139 -id:A070139 - cp's OEIS frontend

A051516 Number of triangles with perimeter n having integer sides and area.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 4, 0, 1, 0, 0, 0, 3, 0, 0, 0, 5, 0, 1, 0, 1, 0, 2, 0, 5, 0, 0, 0, 1, 0, 1, 0, 4, 0, 0, 0, 8, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 5, 0, 6, 0, 5, 0, 0, 0, 2, 0, 0, 0, 12, 0, 1, 0

Offset: 1

David W. Wilson

nonn

No such triangles with odd perimeter exist.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Seiichi Manyama)
Eric Weisstein's World of Mathematics, Heronian Triangle.

Cf. A024153, A070139.

Table[Sum[Sum[(1 - Ceiling[Sqrt[(n/2) (n/2 - i) (n/2 - k) (i + k - n/2)]] + Floor[Sqrt[(n/2) (n/2 - i) (n/2 - k) (i + k - n/2)]])*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 11 2019 *)

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - ceiling(m) + floor(m)) * sign(floor((i+k)/(n-i-k+1))), where m = sqrt((n/2)*(n/2-i)*(n/2-k)*(i+k-n/2)). - Wesley Ivan Hurt, May 11 2019

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

Original entry on oeis.org

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

Clark Kimberling

nonn

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

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.

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

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

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

A070204 Number of isosceles integer triangles with perimeter n having integral inradius.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A070201(n) - A070203(n).

Seiichi Manyama, Table of n, a(n) for n = 1..5000
Mohammad K. Azarian, Solution to Problem S125: Circumradius and Inradius, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32.
Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Isosceles Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A070139, A059169.

A070145 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an isosceles integer triangle with integer area.

Original entry on oeis.org

39, 52, 269, 372, 375, 862, 957, 972, 1204, 1955, 1971, 1988, 2790, 2796, 3818, 5374, 6522, 6880, 6881, 6921, 7234, 7310, 7341, 7360, 9198, 9207, 10272, 14506, 15101, 15177, 15237, 21289, 21493, 21540, 21552, 21589

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(1)=39: [A070080(39), A070081(39), A070082(39)] = [5=5<6], area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore A070086(39)=area=4*3=12.

Eric Weisstein's World of Mathematics, Heronian Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A070142, A070144, A070139.

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A051516 Number of triangles with perimeter n having integer sides and area.

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

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A070204 Number of isosceles integer triangles with perimeter n having integral inradius.

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A070145 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an isosceles integer triangle with integer area.

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