OFFSET
1,1
COMMENTS
The triples of sides (a,b,c) with a < b < c are in nondecreasing order of middle side, and if middle sides coincide then by increasing order of the largest side. This sequence lists the c's.
Equivalently, lengths of the largest side c of primitive non-equilateral triangles that have an angle of Pi/3; indeed, this side is opposite to the largest angle C.
Also, solutions c of the Diophantine equation b^2 = a^2 - a*c + c^2 with gcd(a,b) = 1 and a < b.
For the corresponding primitive triples and miscellaneous properties and references, see A335893.
When (a, b, c) is a triple, then (c-a, b, c) is another triple, so every c in the data is twice consecutively present according to the corresponding pair (b, c) (see examples).
As B = Pi/3 and C runs from Pi/3 to 2*Pi/3, sin(C) gets a maximum when C = Pi/2 with sin(C) = 1, hence, from law of sinus, b/sin(B) = c/sin(C), c < b/sin(Pi/3) = b * 2/sqrt(3) < 6*b/5. This bound is used in PARI and Maple programs.
This sequence is not increasing. For example, a(41) = a(42) = 171 for triangle with middle side = 151 while a(43) = a(44) = 168 for triangle with middle side = 157.
REFERENCES
V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-298 p. 124, André Desvigne.
LINKS
Wikipedia, Law of sines.
FORMULA
a(n) = A335893(n, 3).
c satisfies c^2 - a*c + a^2 - b^2 = 0 with gcd(a,b) = 1 and a < b.
EXAMPLE
c = 8 appears twice because:
7^2 = 3^2 - 3*8 + 8^2, with triple (3, 7, 8),
7^2 = 5^2 - 5*8 + 8^2, with triple (5, 7, 8).
c = 96 and c = 99 each appear twice associated with b = 91 because:
91^2 = 11^2 - 11*96 + 96^2, with triple (11, 91, 96),
91^2 = 85^2 - 85*96 + 96^2, with triple (85, 91, 96),
91^2 = 19^2 - 19*99 + 99^2, with triple (19, 91, 99),
91^2 = 80^2 - 80*99 + 99^2, with triple (80, 91, 99).
MAPLE
for b from 3 to 250 by 2 do
for c from b+1 to 6*b/5 do
a := (c - sqrt(4*b^2-3*c^2))/2;
if gcd(a, b)=1 and issqr(4*b^2-3*c^2) then print(c, c); end if;
end do;
end do;
PROG
(PARI) lista(nn) = {forstep(b=1, nn, 2, for(c=b+1, 6*b\5, if (issquare(d=4*b^2 - 3*c^2), my(a = (c - sqrtint(d))/2); if ((denominator(a)==1) && (gcd(a, b) == 1), print(c, ", ", c, ", "); ); ); ); ); } \\ Michel Marcus, Jul 15 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, Jul 10 2020
STATUS
approved