A055595 -id:A055595 - cp's OEIS frontend

A135622 16*Area^2 of integer triangles [A070080(n),A070081(n),A070082(n)].

3, 15, 48, 35, 63, 128, 63, 135, 243, 240, 320, 99, 231, 275, 495, 384, 576, 768, 143, 351, 455, 819, 975, 560, 896, 1008, 1344, 195, 495, 675, 1215, 735, 1575, 1875, 768, 1280, 1536, 2048, 2304, 255, 663, 935, 1683, 1071, 2295, 2499, 2975, 1008, 1728

Offset: 1

Franz Vrabec, Feb 29 2008

			A070080(4)=1, A070081(4)=3, A070082(4)=3, so a(4)=(1+3+3)*(-1+3+3)*(1-3+3)*(1+3-3)=35.

See the formula section for the relationships with A070080, A070081, A070082, A070086.
Cf. A317182 (range of values), A331011 (nonunique values), A331250 (counts triangles by area).
Cf. A316853 (with terms ordered as for A316841), and using this order for other sets of triangles: A046131, A055595, A070786.

a(n)=(u+v+w)*(-u+v+w)*(u-v+w)*(u+v-w), where u=A070080(n), v=A070081(n), w=A070082(n).

A070086(n) = round(sqrt(a(n))/4).

A336272 Length of longest side of a primitive square Heron triangle, i.e., a triangle with relatively prime integer sides and area the square of a positive integer.

Original entry on oeis.org

17, 26, 120, 370, 392, 567, 680, 697, 847, 1066, 1089, 1183, 1233, 1299, 1371, 1448, 1904, 2009, 2169, 2176, 2281, 2307, 2535, 2600, 2619, 2785, 2845, 2993, 3150, 3370, 3825, 3944, 3983, 4035, 4095, 4290, 4706, 4760, 4879, 4905, 5655, 5811, 5835, 6137, 6375, 6570, 6936, 7202, 7913, 7995

Offset: 1

James R. Buddenhagen, Jul 15 2020

nonn

The triangle [a(23)=2535, 2329, 544] with gcd(2329, 544) = 17 is the first square Heron triangle for which the 3 sides [i, j, k] are not pairwise coprime, i.e., max(gcd(i,j), gcd(i,k), gcd(j,k)) > 1, but gcd(i,j,k) = 1. Are there more square Heron triangles with this property? - Hugo Pfoertner, Jul 18 2020
There are other square Heron triangles with this property, e.g. [a(31)=3825, 2704, 1921] with gcd(1921, 3825) = 17; [a(??)=41460721, 38639097, 17536520] with gcd(38639097, 17536520) = 41; [a(??)=153915025, 139641489, 25224736] with gcd(25224736, 153915025) = 17; and [a(??)=4325561361, 3459908000, 1430190961] with gcd(3459908000, 1430190961) = 73. - James R. Buddenhagen, Jul 20 2020
Terms are given with multiplicity, e.g. if there are two primitive square Heron triangles with equal longest sides, that longest side is listed as a term of the sequence twice (this is very rare). - James R. Buddenhagen, Jul 21 2020

			17 is in the sequence because the triangle with sides [17, 10, 9] has longest side 17 and area 6^2, the square of a positive integer; 26 is in the sequence because the triangle with sides [26, 25, 3] has longest side 26 and has area 6^2, the square of a positive integer.
Triangles with sides [a, b, c] corresponding to the first 8 terms of this sequence are:  [17, 10, 9], [26, 25, 3], [120, 113, 17], [370, 357, 41], [392, 353, 255], [567, 424, 305], [680, 441, 337], [697, 657, 104].

Hugo Pfoertner, Table of n, a(n) for n = 1..79
Sascha Kurz, On the generation of Heronian triangles, arXiv:1401.6150 [math.NT], 11 Jan 2014.
Sascha Kurz, On the generation of Heronian triangles, Serdica Journal of Computing Vol. 2 (2008), Pages 181-196.
Hugo Pfoertner, List of triangle sides, 20000 > i > j > k.

Cf. A046128, A046129, A046130, A046131, A055592, A055593, A055594, A055595.

# find all square Heron triangles whose longest side is between small and big
small:=1: big:=700:
A336272:=[]:triangles:=[]:
areasq16:=(a+b+c)*(a+b-c)*(a-b+c)*(-a+b+c):
# a>=b>=c
for a from small to big do:
  for b from ceil((a+1)/2) to a do:
    for c from a-b+1 to b do:
      if issqr(areasq16) and issqr(sqrt(areasq16)) and igcd(a,b,c)=1 then
        A336272:=[op(A336272),a]:
        triangles:=[op(triangles),[a,b,c]]:
      end if:
    od:
  od:
od: A336272;triangles;

for(a=1,1200,for(b=ceil((a+1)/2),a,for(c=a-b+1,b,if(gcd([a,b,c])==1,if(ispower((a+b+c)*(a+b-c)*(a-b+c)*(b+c-a),4),print1(a,", ")))))) \\ Hugo Pfoertner, Jul 18 2020

a(42)-a(50) from Hugo Pfoertner, Jul 18 2020

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A135622 16*Area^2 of integer triangles [A070080(n),A070081(n),A070082(n)].

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A336272 Length of longest side of a primitive square Heron triangle, i.e., a triangle with relatively prime integer sides and area the square of a positive integer.

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