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%I #31 Jun 25 2021 03:00:13
%S 8,8,15,15,21,21,35,35,40,40,48,48,55,55,65,65,77,77,80,80,91,91,96,
%T 96,99,99,112,112,117,117,119,119,133,133,143,143,153,153,160,160,171,
%U 171,168,168,187,187,176,176,209,209,207,207,221,221,224,224,225,225
%N Largest side of integer-sided primitive triangles whose angles A < B < C are in arithmetic order.
%C 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.
%C 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.
%C Also, solutions c of the Diophantine equation b^2 = a^2 - a*c + c^2 with gcd(a,b) = 1 and a < b.
%C For the corresponding primitive triples and miscellaneous properties and references, see A335893.
%C 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).
%C 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.
%C 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.
%D V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-298 p. 124, André Desvigne.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Law_of_sines">Law of sines</a>.
%F a(n) = A335893(n, 3).
%F c satisfies c^2 - a*c + a^2 - b^2 = 0 with gcd(a,b) = 1 and a < b.
%e c = 8 appears twice because:
%e 7^2 = 3^2 - 3*8 + 8^2, with triple (3, 7, 8),
%e 7^2 = 5^2 - 5*8 + 8^2, with triple (5, 7, 8).
%e c = 96 and c = 99 each appear twice associated with b = 91 because:
%e 91^2 = 11^2 - 11*96 + 96^2, with triple (11, 91, 96),
%e 91^2 = 85^2 - 85*96 + 96^2, with triple (85, 91, 96),
%e 91^2 = 19^2 - 19*99 + 99^2, with triple (19, 91, 99),
%e 91^2 = 80^2 - 80*99 + 99^2, with triple (80, 91, 99).
%p for b from 3 to 250 by 2 do
%p for c from b+1 to 6*b/5 do
%p a := (c - sqrt(4*b^2-3*c^2))/2;
%p if gcd(a,b)=1 and issqr(4*b^2-3*c^2) then print(c,c); end if;
%p end do;
%p end do;
%o (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
%Y Cf. A089025 (terms in increasing order without repetition).
%Y Cf. A335893 (triples), A335894 (smallest side), A335895 (middle side), this sequence (largest side), A335897 (perimeter).
%K nonn
%O 1,1
%A _Bernard Schott_, Jul 10 2020
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