A375748 - OEIS

%I #7 Oct 09 2024 18:31:27

%S 2,7,15,24,62,55,132,120,191,275,311,300,722,710,703,655,1107,1027,

%T 1500,1483,1890,1823,1806,1746,4520,4315,4250,4156,4133,4027,3980,

%U 3896,6663,6625,6497,6240,9083,9030,8786,8730,12403,11990,11918,11885,11789,11302,11210,11138,27560

%N a(n) is the smallest possible side x in a family of triangles with integer sides x, y < x, x-y < z < x+y, such that exactly n pairs of triangles with equal area exist in this family.

%H IBM Research, <a href="https://research.ibm.com/haifa/ponderthis/challenges/September2024.html">Sibling triangles</a>, Ponder This Challenge September 2024, asked for families with exactly 50 pairs.

%H Hugo Pfoertner, <a href="/A375748/a375748.txt">List of a(n) and A375749(n)</a> for n=0..71.

%e   n   x=a(n)

%e   |   |   y=A375749(n)

%e   |   |      pairs of sides z leading to equal areas

%e   0   2   1  only 1 triangle, no pair

%e   1   7   4  [7,9]

%e   2  15  10  [11,23], [17,19]

%e   3  24  23  [19,43], [23,41], [29,37]

%e   4  62  41  [45,95], [49,93], [59,87], [67,81]

%o (PARI) A(a,b,c) = (a+b+c)*(a+b-c)*(a-b+c)*(b+c-a); \\ squared area * 16

%o check(a,b) = {my(F=List()); for(c=a-b+1, a+b-1, listput(F, A(a,b,c))); F=vecsort(F); my(p=F[1], pc=1, mf=0); for(k=2, #F, if(F[k]==p, pc++; mf++, pc=1; p=F[k])); mf};

%o \\ returns [a(n), A375749(n)]

%o a375748_9(n) = for(a=2, oo, for(b=1, a-1, if(check(a,b)==n, return([a,b]))))

%Y A375749 gives the corresponding side y.

%K nonn

%O 0,1

%A _Hugo Pfoertner_, Sep 09 2024