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-11, 21, 24, 28, 40, 52, 61, 157, 76, 85, 96, 117, 120, 132, 181, 213, 237, 376, 388, 397, 132, 156, 160, 189, 204, 205, 216, 237, 253, 285, 288, 309, 316, 336, 349, 405, 412, 421, 453, 460, 469, 472, 517, 544, 565, 616, 628, 685, 717, 741, 1084, 1093, 1104, 1125, 1128, 1140

root = {-11, 21, 24, 28};

triples = Subsets[root, {3}];

a = {root};

Do[

ng = Table[Total@t + 2 Sqrt@Total[Times @@@ Subsets[t, {2}]], {t, triples}];

AppendTo[a, Sort@ng];

triples = Join @@ Table[{t, r} = tr; Table[Append[p, r], {p, Subsets[t, {2}]}], {tr, Transpose@{triples, ng}}]

, {k, 3}];

Flatten@a (* Andrey Zabolotskiy, May 29 2022 *)

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Terms a(28) and beyond from Andrey Zabolotskiy, May 29 2022

J. Bourgain and A. Kontorovich, <a href="http://arxiv.org/abs/1205.4416">On the Strong Density Conjecture for Integral Apollonian Circle Packings</a>, arXiv:1205.4416 [math.NT], 2012-2013. See figure 1.

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Apollonius' s and Descartes' s Theorems say that, given three mutually tangent circles of curvatures a, b, c, there are exactly two circles tangent to all three, and their curvatures are a + b + c +/- 2sqrt2*sqrt(ab + ac + bc). (Here negative curvature of one of the two circles means that the three circles are inscribed in it.)

Fuchs (2009) says "An Apollonian circle packing ... is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well." That is because if a + b + c - 2sqrt2s*qrt(ab + ac + bc) is an integer, then so is a + b + c + 2sqrt2*sqrt(ab + ac + bc).

The 1st -generation curvatures are -11, 21, 24, 28, the 2nd are 40, 52, 61, 157, and the 3rd are 76, 85, 96, 117, 120, 132, 181, 213, 237, 376, 388, 397. The 4th generation begins 132, 156, 160, 189, 204, 205, 216, ....

As 21 + 24 + 28 +/- 2sqrt2*sqrt(21*24 + 21*28 + 24*28) = 157 or -11, the sequence begins -11, 21, 24, 28, ... and 157 is in it.

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