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

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, 16, 16, 17, 18, 18, 18, 19, 20, 20, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 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

Offset

1,1

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Maple

F:= proc(a)
  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))
end proc:
seq(a$nops(F(a)), a=1..40); # Robert Israel, Dec 18 2024

Mathematica

(* Warning: this code is incorrect, as it imposes a limit b <= 900 *)
z8 = 900; z9 = 250; z7 = 200;
k = 1/4; c[a_, b_] := Sqrt[a^2 + b^2 + k*a*b];
d[a_, b_] := If[IntegerQ[c[a, b]], {a, b, c[a, b]}, 0]
t[a_] := Table[d[a, b], {b, a, z8}]
u[n_] := Delete[t[n], Position[t[n], 0]]
Table[u[n], {n, 1, 15}]
t = Table[u[n], {n, 1, z8}];
Flatten[Position[t, {}]]
u = Flatten[Delete[t, Position[t, {}]]];
x[n_] := u[[3 n - 2]];
Table[x[n], {n, 1, z7}]  (* A196259 *)
y[n_] := u[[3 n - 1]];
Table[y[n], {n, 1, z7}]  (* A196260 *)
z[n_] := u[[3 n]];
Table[z[n], {n, 1, z7}]  (* A196261 *)
x1[n_] := If[GCD[x[n], y[n], z[n]] == 1, x[n], 0]
y1[n_] := If[GCD[x[n], y[n], z[n]] == 1, y[n], 0]
z1[n_] := If[GCD[x[n], y[n], z[n]] == 1, z[n], 0]
f = Table[x1[n], {n, 1, z9}];
x2 = Delete[f, Position[f, 0]]  (* A196262 *)
g = Table[y1[n], {n, 1, z9}];
y2 = Delete[g, Position[g, 0]]  (* A196263 *)
h = Table[z1[n], {n, 1, z9}];
z2 = Delete[h, Position[h, 0]]  (* A196264 *)

Crossrefs

Cf. A195770, A196260, A196261, A196262.

Sequence in context: A276523 A244320 A084616 * A214880 A071193 A071192

Adjacent sequences: A196256 A196257 A196258 * A196260 A196261 A196262

Keyword

nonn

Author

Clark Kimberling, Sep 30 2011

Extensions

Corrected by Robert Israel, Dec 18 2024

Status

approved