A265495 a(n) is the smallest k > 0 for which there exists a root quad (-k,x,y,z) such that some bend is first repeated in the n-th generation of descendants (by Descartes reflection).

1, 2, 7, 6, 14, 29

Offset

0,2

Comments

Perhaps a(0) should be 0, for the quad (0,0,1,1).

Functions were written in the statistical language R to generate root quads and to generate successive generations of descendants. The n-th generation (n >= 1) contains 4*3^(n-1) quads.

Links

Table of n, a(n) for n=0..5.

R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks and C. H. Yan, Apollonian Circle Packings: Number Theory, arXiv:math/0009113 [math.NT], 2000-2003.

R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks and C. H. Yan, Apollonian Circle Packings: Number Theory, J. Number Theory, 100 (2003), 1-45.

Example

For n = 0,1,2,3,4,5, qualifying root quads are (-1,2,2,3), (-2,3,6,7), (-7,12,17,20), (-6,11,14,15), (-14,19,54,55), (-29,55,60,70). E.g., for n=3, the bend 71 appears in both the second and third generations, in the quads (-6,14,35,71) and (-6,11,42,71).

Crossrefs

Cf. A042944, A045864, A261410.

Sequence in context: A154200 A089417 A168205 * A338041 A365007 A082017

Adjacent sequences: A265492 A265493 A265494 * A265496 A265497 A265498

Keyword

nonn,more

Author

Colin Mallows, Dec 09 2015

Status

approved