A261654 Lead almost-Pythagorean triples generated by primitive Pythagorean triples of the form (2i-1, 2i^2-2i, 2i^2-2i+1), i >= 2.

4, 7, 8, 6, 17, 18, 8, 31, 32, 10, 49, 50, 12, 71, 72, 14, 97, 98, 16, 127, 128, 18, 161, 162, 20, 199, 200, 22, 241, 242, 24, 287, 288, 26, 337, 338, 28, 391, 392, 30, 449, 450, 32, 511, 512, 34, 577, 578

Offset

1,1

Comments

A set of ordered triple (x,y,z) that satisfies the equation x^2 + y^2 = z^2 + 1 is called an almost-Pythagorean triple (APT).

The triples (x,y,z)=[(2i-1)k+1,(2i^2-2i)k+(2i-1),(2i^2-2i+1)k+(2i-1)] and (x',y',z')=[(2i-1)k+(2i-2),(2i^2-2i)k+(2i^2-4i+1),(2i^2-2i+1)k+(2i^2-4i+2)] are APTs for all integers k and i >= 2.

Note that in terms of components, (x,y,z) < (x',y',z').

Setting k=1 in the first expression gives the terms of this sequence.

Links

Table of n, a(n) for n=1..48.

John Rafael M. Antalan, Mark D. Tomenes, A Note on Generating Almost Pythagorean Triples, arXiv:1508.07562 [math.NT], 2015.

O. Frink, Almost Pythagorean Triples, Mathematics Magazine, Vol.60, No.4, (1987), pp.234-236.

Formula

(x,y,z) = [(2i-1)k+1,(2i^2-2i)k+(2i-1),(2i^2-2i+1)k+(2i-1)], with i>=2 and k=1.

Example

When k=1 and i=2 the formula for (x,y,z) gives the Lead APT (4,7,8).
First rows are:
   4,  7,  8;
   6, 17, 18;
   8, 31, 32;
  10, 49, 50;
  12, 71, 72;
  14, 97, 98;
  ...

Mathematica

xyz[i_] := {2i, 2i^2-1, 2i^2};
Array[xyz, 16, 2] // Flatten (* Jean-François Alcover, Feb 02 2019 *)

Prog

(PARI) tabf(nn) = for (i=2, nn, print(2*i, ", ", 2*i^2-1, ", ", 2*i^2)); \\ Michel Marcus, Aug 31 2015

Crossrefs

For the 3 columns, cf. A005843, A056220, A001105.

Sequence in context: A115021 A200367 A272490 * A332504 A121488 A115291

Adjacent sequences: A261651 A261652 A261653 * A261655 A261656 A261657

Keyword

nonn,tabf

Author

John Rafael M. Antalan, Aug 30 2015

Status

approved