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A261654 Lead almost-Pythagorean triples generated by primitive Pythagorean triples of the form (2i-1, 2i^2-2i, 2i^2-2i+1), i >= 2. 0
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 (list; graph; refs; listen; history; text; internal format)
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;

  ...

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 * A121488 A115291 A108615

Adjacent sequences:  A261651 A261652 A261653 * A261655 A261656 A261657

KEYWORD

nonn,tabf

AUTHOR

John Rafael M. Antalan, Aug 30 2015

STATUS

approved

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Last modified January 17 04:06 EST 2019. Contains 319206 sequences. (Running on oeis4.)