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A196259 Positive integers a in (1/4)-Pythagorean triples (a,b,c) satisfying a<=b, in order of increasing a and then increasing b.

%I A196259 #21 Dec 20 2024 11:08:17
%S A196259 2,4,4,5,6,6,7,7,8,8,9,10,10,10,11,12,12,12,12,13,14,14,14,15,15,15,
%T A196259 16,16,17,18,18,18,19,20,20,20,20,20,20,21,21,22,22,22,22,23,23,24,24,
%U A196259 24,24,25,25,25,26,26,26,27,28,28,28,28,28,29,30,30,30,30,30,30,31,31,32,32,32,32,32
%N A196259 Positive integers a in (1/4)-Pythagorean triples (a,b,c) satisfying a<=b, in order of increasing a and then increasing b.
%C A196259 See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.
%H A196259 Robert Israel, <a href="/A196259/b196259.txt">Table of n, a(n) for n = 1..10000</a>
%p A196259 F:= proc(a)
%p A196259   sort(select(t -> subs(t,b) >= a and subs(t,c) > 0, [isolve](4*a^2 + 4*b^2 + a*b = 4*c^2)),(s,t) -> subs(s,b) <= subs(t,b))
%p A196259 end proc:
%p A196259 seq(a$nops(F(a)), a=1..40); # _Robert Israel_, Dec 18 2024
%t A196259 (* Warning: this code is incorrect, as it imposes a limit b <= 900 *)
%t A196259 z8 = 900; z9 = 250; z7 = 200;
%t A196259 k = 1/4; c[a_, b_] := Sqrt[a^2 + b^2 + k*a*b];
%t A196259 d[a_, b_] := If[IntegerQ[c[a, b]], {a, b, c[a, b]}, 0]
%t A196259 t[a_] := Table[d[a, b], {b, a, z8}]
%t A196259 u[n_] := Delete[t[n], Position[t[n], 0]]
%t A196259 Table[u[n], {n, 1, 15}]
%t A196259 t = Table[u[n], {n, 1, z8}];
%t A196259 Flatten[Position[t, {}]]
%t A196259 u = Flatten[Delete[t, Position[t, {}]]];
%t A196259 x[n_] := u[[3 n - 2]];
%t A196259 Table[x[n], {n, 1, z7}]  (* A196259 *)
%t A196259 y[n_] := u[[3 n - 1]];
%t A196259 Table[y[n], {n, 1, z7}]  (* A196260 *)
%t A196259 z[n_] := u[[3 n]];
%t A196259 Table[z[n], {n, 1, z7}]  (* A196261 *)
%t A196259 x1[n_] := If[GCD[x[n], y[n], z[n]] == 1, x[n], 0]
%t A196259 y1[n_] := If[GCD[x[n], y[n], z[n]] == 1, y[n], 0]
%t A196259 z1[n_] := If[GCD[x[n], y[n], z[n]] == 1, z[n], 0]
%t A196259 f = Table[x1[n], {n, 1, z9}];
%t A196259 x2 = Delete[f, Position[f, 0]]  (* A196262 *)
%t A196259 g = Table[y1[n], {n, 1, z9}];
%t A196259 y2 = Delete[g, Position[g, 0]]  (* A196263 *)
%t A196259 h = Table[z1[n], {n, 1, z9}];
%t A196259 z2 = Delete[h, Position[h, 0]]  (* A196264 *)
%Y A196259 Cf. A195770, A196260, A196261, A196262.
%K A196259 nonn
%O A196259 1,1
%A A196259 _Clark Kimberling_, Sep 30 2011
%E A196259 Corrected by _Robert Israel_, Dec 18 2024