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Positive integers a in primitive (1/4)-Pythagorean triples (a,b,c) satisfying a<=b, in order of increasing a and then increasing b.
6

%I #17 Dec 26 2024 20:00:59

%S 2,4,5,6,7,7,9,10,11,12,13,15,15,17,19,20,20,22,22,23,23,25,25,26,28,

%T 29,31,31,32,32,32,33,34,36,37,38,38,39,39,41,41,43,44,44,46,47,47,50,

%U 52,52,52,53,54,55,55,55,55,57,58,58,59,60,61,62,63,64,64,64,64,65,65,65,67,68,68,68

%N Positive integers a in primitive (1/4)-Pythagorean triples (a,b,c) satisfying a<=b, in order of increasing a and then increasing b.

%C See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.

%H Robert Israel, <a href="/A196262/b196262.txt">Table of n, a(n) for n = 1..10000</a>

%e Primitive (1/4)-Pythagorean triples a,b,c where c^2=a^2+b^2+(1/4)*a*b:

%e 2, 2, 3

%e 4, 15, 16

%e 5, 32, 33

%e 6, 70, 71

%e 7, 20, 22

%e 7,192,193

%e 9, 44, 46

%e 10, 26, 29

%e 11, 20, 24

%e 12, 17, 22

%p F:= proc(a)

%p sort(select(t -> subs(t, b) >= a and subs(t, c) > 0 and igcd(a, subs(t,b),subs(t,c)) = 1, [isolve](4*a^2 + 4*b^2 + a*b = 4*c^2)), (s, t) -> subs(s, b) <= subs(t, b))

%p end proc:

%p seq(a$nops(F(a)),a=1..100);# _Robert Israel_, Dec 20 2024

%t (See A196259.)

%Y Cf. A195770, A196259, A196263, A196264.

%K nonn

%O 1,1

%A _Clark Kimberling_, Sep 30 2011

%E Corrected by _Robert Israel_, Dec 20 2024