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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
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
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
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Sep 09 2024
STATUS
approved