A375748 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.

2, 7, 15, 24, 62, 55, 132, 120, 191, 275, 311, 300, 722, 710, 703, 655, 1107, 1027, 1500, 1483, 1890, 1823, 1806, 1746, 4520, 4315, 4250, 4156, 4133, 4027, 3980, 3896, 6663, 6625, 6497, 6240, 9083, 9030, 8786, 8730, 12403, 11990, 11918, 11885, 11789, 11302, 11210, 11138, 27560

Offset

0,1

Links

Table of n, a(n) for n=0..48.

IBM Research, Sibling triangles, Ponder This Challenge September 2024, asked for families with exactly 50 pairs.

Hugo Pfoertner, List of a(n) and A375749(n) for n=0..71.

Example

  n   x=a(n)
  |   |   y=A375749(n)
  |   |      pairs of sides z leading to equal areas
  0   2   1  only 1 triangle, no pair
  1   7   4  [7,9]
  2  15  10  [11,23], [17,19]
  3  24  23  [19,43], [23,41], [29,37]
  4  62  41  [45,95], [49,93], [59,87], [67,81]

Prog

(PARI) A(a, b, c) = (a+b+c)*(a+b-c)*(a-b+c)*(b+c-a); \\ squared area * 16
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};
\\ returns [a(n), A375749(n)]
a375748_9(n) = for(a=2, oo, for(b=1, a-1, if(check(a, b)==n, return([a, b]))))

Crossrefs

A375749 gives the corresponding side y.

Sequence in context: A132746 A252475 A350043 * A167543 A332495 A184976

Adjacent sequences: A375745 A375746 A375747 * A375749 A375750 A375751

Keyword

nonn

Author

Hugo Pfoertner, Sep 09 2024

Status

approved