A060782 Degree of the curve C(n) relative to a triangle ABC with side lengths a, b and c, given by (x^n- b^n)(y^n-c^n)(z^n-a^n) = (x^n-c^n)(y^n-a^n)(z^n-b^n) where x, y and z denote the distances from the variable point to vertices A, B and C respectively.

14, 3, 46, 7, 78, 11

Offset

1,1

Comments

a(n) is also the degree of C(-n).

Links

Table of n, a(n) for n=1..6.

A. P. Hatzipolakis, F. van Lamoen, B. Wolk and Paul Yiu, Concurrency of four Euler lines, Forum Geometricorum 1 (2001) 59 - 68.

Formula

It is known that a(n) = 2n-1 for n even; a(1) = 14; a(3) = 46; and it is conjectured that a(n) = 16n-2 for n odd.

Example

C(2) is a cubic curve, the well-known Neuberg cubic of a triangle, so a(2)=3.

Crossrefs

Sequence in context: A040190 A317314 A163647 * A182431 A182440 A040187

Adjacent sequences: A060779 A060780 A060781 * A060783 A060784 A060785

Keyword

nonn,more

Author

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Apr 28 2001

Extensions

Better description from Floor van Lamoen, Jul 10 2001

a(6) from Sean A. Irvine, Jan 01 2023

Status

approved