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A343067 Perimeter of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

%I A343067 #29 May 13 2022 05:52:52
%S A343067 15,28,45,66,91,120,40,153,190,231,276,84,325,378,435,496,144,77,561,
%T A343067 630,703,104,780,220,861,946,1035,1128,312,1225,170,1326,1431,1540,
%U A343067 126,420,209,1653,1770,1891,2016,544,2145,2278,299,2415,2556,198,684,2701,350,2850,3003,3160
%N A343067 Perimeter of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.
%C A343067 The triples (a, b, c) are listed in increasing order of side a, and if sides a coincide then in increasing order of side b.
%C A343067 This sequence is nonincreasing: a(7) = 40 < a(6) = 120.
%C A343067 If in triangle ABC, B = 2*C, then the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.
%C A343067 As the metric relation is equivalent to a = m^2 - k^2, b = m*k, c = k^2, with gcd(m,k) = 1 and k < m < 2k, so all terms are of the form m^2 + m*k = m * (m+k) with gcd(m,k) = 1 and k < m < 2k. These perimeters are in increasing order in A106499.
%C A343067 For the corresponding primitive triples and miscellaneous properties and references, see A343063.
%F A343067 a(n) = A343063(n, 1) + A343063(n, 2) + A343063(n, 3).
%F A343067 a(n) = A343064(n) + A343065(n) + A343066(n).
%e A343067 According to inequalities between a, b, c, there exist 3 types of such triangles:
%e A343067 a(1) = 15 with c < a < b for the first triple (5, 6, 4);
%e A343067 a(7) = 40 with c < b < a for the seventh triple (16, 15, 9);
%e A343067 a(8) = 153 with a < c < b for the eighth triple (17, 72, 64).
%p A343067 for a from 2 to 100 do
%p A343067 for c from 3 to floor(a^2/2) do
%p A343067 d := c*(a+c);
%p A343067 if issqr(d) and igcd(a, sqrt(d), c)=1 and abs(a-c)<sqrt(d) and sqrt(d)<a+c then print(a+sqrt(d)+c); end if;
%p A343067 end do;
%p A343067 end do;
%o A343067 (PARI) lista(nn) = {for (a = 2, nn, for (c = 3, a^2\2, my(d = c*(a+c)); if (issquare(d) && (gcd([a, sqrtint(d), c])==1) && (abs(a-c)<sqrtint(d)) && (sqrtint(d)<a+c),  print1(a+sqrtint(d)+c ", "));););} \\ _Michel Marcus_, May 12 2022
%Y A343067 Cf. A335897 (similar for A < B < C in arithmetic progression).
%Y A343067 Cf. A343063 (triples), A343064 (side a), A343065 (side b), A343066 (side c), A106499 (perimeters in increasing order).
%K A343067 nonn
%O A343067 1,1
%A A343067 _Bernard Schott_, Apr 15 2021