A195778 Positive integers a for which there is a (-1)-Pythagorean triple (a,b,c) satisfying a<=b.

1, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..72.

Example

First five (-1)-Pythagorean triples (A195778):
(1,1,1), (2,2,2), (3,3,3), (3,8,7), (4,4,4).
First five primitive (-1)-Pythagorean triples (A195796):
(1,1,1), (3,8,7), (5,8,7), (5,21,19), (7,15,13).

Mathematica

z8 = 800; z9 = 400; z7 = 100;
k = -1; 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}]  (* A195778 *)
y[n_] := u[[3 n - 1]];
Table[y[n], {n, 1, z7}]  (* A195794 *)
z[n_] := u[[3 n]];
Table[z[n], {n, 1, z7}]  (* A195795 *)
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]] (* A195796 *)
g = Table[y1[n], {n, 1, z9}];
y2 = Delete[g, Position[g, 0]] (* A195862 *)
h = Table[z1[n], {n, 1, z9}];
z2 = Delete[h, Position[h, 0]] (* A195863 *)

Crossrefs

Cf. A195770, A009004, A195794, A195795, A195796, A195862, A195863.

Sequence in context: A083036 A241920 A202305 * A165706 A242453 A241151

Adjacent sequences: A195775 A195776 A195777 * A195779 A195780 A195781

Keyword

nonn

Author

Clark Kimberling, Sep 25 2011

Status

approved