cp's OEIS Frontend

A237576 Smallest integer areas of integer-sided triangles such that the perimeter equals n times the smallest side.

%I A237576 #18 Feb 25 2025 04:50:22
%S A237576 0,0,0,6,60,30,210,24,84,60,198,330,1716,546,2730,252,4080,36,5814,
%T A237576 210,7980,2310,10626,924,1380,1248,90,4914,4176,6090,26970,480,32736,
%U A237576 1224,39270,1938,46620,2394,54834,4560,63960,4620,74046,19866,85140,22770,97290
%N A237576 Smallest integer areas of integer-sided triangles such that the perimeter equals n times the smallest side.
%C A237576 The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2.
%C A237576 The sequence a(n) is the union of four subsequences A, B, C and D where:
%C A237576 A is the subsequence with areas 60, 210, 1716, 2730, 4080, 5814, 7980, 10626, ... where n is odd, and the corresponding sides are of the form (4k, 4k^2-1, 4k^2+1) with areas 2k(4k^2-1) for k = 2, 3, 6, 7, 8, 9, 11, ... These areas are in the sequence A069072 (areas of primitive Pythagorean triangles whose odd sides differ by 2).
%C A237576 B is the subsequence with areas 6, 30, 84, 330, 546, 2310, 4914, 6090, ... where n is even, and the corresponding sides are of the form (2k+1, 2k(k+1), 2k(k+1)+1) with areas k(k+1)(2k+1) for k = 1, 2, 3, 5, 6, 7, 10, 13, 14, ... These areas are in the sequence A055112 (Areas of Pythagorean triangles (a, b, c) with c = b+1).
%C A237576 C is the subsequence with areas 84, 198, 1380, 4176, ... where n is odd but the areas are not Pythagorean triangles.
%C A237576 D is the subsequence with areas 24, 60, 210, 924, 1248, 480, 1224, 1938, ... where n is even but the areas are not Pythagorean triangles.
%C A237576 The triangles with the same areas are not unique; for example:
%C A237576 (8, 15, 17) and (6, 25, 29) => A = 60; the first is a Pythagorean triangle, the second is not.
%C A237576 (12, 35, 37) and (7, 65, 68) => A = 210; the first is a Pythagorean triangle, the second is not.
%C A237576 The following table gives the first values (n, A, p, a, b, c) where A is the area of the triangles, p is the perimeter and a, b, c are the sides.
%C A237576 +----+------+-------------+----+-----+-----+
%C A237576 |  n |   A  |      p      |  a |  b  |  c  |
%C A237576 +----+------+-------------+----+-----+-----+
%C A237576 |  4 |    6 |  12 = 4*3   |  3 |   4 |   5 |
%C A237576 |  5 |   60 |  40 = 5*8   |  8 |  15 |  17 |
%C A237576 |  6 |   30 |  30 = 6*5   |  5 |  12 |  13 |
%C A237576 |  7 |  210 |  84 = 7*12  | 12 |  35 |  37 |
%C A237576 |  8 |   24 |  32 = 8*4   |  4 |  13 |  15 |
%C A237576 |  9 |   84 |  72 = 9*8   |  8 |  29 |  35 |
%C A237576 | 10 |   60 |  60 = 10*6  |  6 |  25 |  29 |
%C A237576 | 11 |  198 | 132 = 11*12 | 12 |  55 |  65 |
%C A237576 | 12 |  330 | 132 = 12*11 | 11 |  60 |  61 |
%C A237576 | 13 | 1716 | 312 = 13*24 | 24 | 143 | 145 |
%C A237576 | 14 |  546 | 182 = 14*13 | 13 |  84 |  85 |
%C A237576 | 15 | 2730 | 420 = 15*28 | 28 | 195 | 197 |
%C A237576 +----+------+-------------+----+-----+-----+
%p A237576 with(numtheory):nn:=600:for n from 4 to 50 do: ii:=0:for a from 1
%p A237576   to nn while(ii=0) do: for b from a to nn while(ii=0) do: for c from b to nn while(ii=0) do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c): if x>0 then x0:= sqrt(x):else fi:if x0=floor(x0) and 2*p=n*a then ii:=1:printf ( "%d %d %d %d %d \n",n,x0,a,b,c):else fi:od:od:od:od:
%t A237576 nn=600;lst={};Do[k=0;Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s (s-a) (s-b) (s-c);If[0<area2&&k==0&&IntegerQ[Sqrt[area2]]&&2*s==n*Min[a,b,c], AppendTo[lst,Sqrt[area2]];k=1]],{a,nn},{b,a},{c,b}],{n,4,45}];Union[lst]
%Y A237576 Cf. A188158.
%K A237576 nonn
%O A237576 1,4
%A A237576 _Michel Lagneau_, Feb 09 2014