%I #42 Feb 02 2019 10:42:33

%S 4,7,8,6,17,18,8,31,32,10,49,50,12,71,72,14,97,98,16,127,128,18,161,

%T 162,20,199,200,22,241,242,24,287,288,26,337,338,28,391,392,30,449,

%U 450,32,511,512,34,577,578

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

%C 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).

%C 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.

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

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

%H John Rafael M. Antalan, Mark D. Tomenes, <a href="http://arxiv.org/abs/1508.07562">A Note on Generating Almost Pythagorean Triples</a>, arXiv:1508.07562 [math.NT], 2015.

%H O. Frink, <a href="http://www.jstor.org/stable/2689346">Almost Pythagorean Triples</a>, Mathematics Magazine, Vol.60, No.4, (1987), pp.234-236.

%F (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.

%e When k=1 and i=2 the formula for (x,y,z) gives the Lead APT (4,7,8).

%e First rows are:

%e 4, 7, 8;

%e 6, 17, 18;

%e 8, 31, 32;

%e 10, 49, 50;

%e 12, 71, 72;

%e 14, 97, 98;

%e ...

%t xyz[i_] := {2i, 2i^2-1, 2i^2};

%t Array[xyz, 16, 2] // Flatten (* _Jean-François Alcover_, Feb 02 2019 *)

%o (PARI) tabf(nn) = for (i=2, nn, print(2*i, ", ", 2*i^2-1, ", ", 2*i^2)); \\ _Michel Marcus_, Aug 31 2015

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

%K nonn,tabf

%O 1,1

%A _John Rafael M. Antalan_, Aug 30 2015