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

Author

Colin Mallows, Dec 09 2015

Keywords

• nonn
• more

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.

Examples

			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).
		

Links

• 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.

Crossrefs

Cf. A042944, A045864, A261410.