

A261654


Lead almostPythagorean triples generated by primitive Pythagorean triples of the form (2i1, 2i^22i, 2i^22i+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
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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 almostPythagorean triple (APT).
The triples (x,y,z)=[(2i1)k+1,(2i^22i)k+(2i1),(2i^22i+1)k+(2i1)] and (x',y',z')=[(2i1)k+(2i2),(2i^22i)k+(2i^24i+1),(2i^22i+1)k+(2i^24i+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.234236.


FORMULA

(x,y,z) = [(2i1)k+1,(2i^22i)k+(2i1),(2i^22i+1)k+(2i1)], 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^21, ", ", 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



