A102762 Curvatures of (largest) kissing circles along the circumference, starting with curvature = -1 and 2.

-1, 2, 2, 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403, 2502, 2603, 2706, 2811, 2918, 3027, 3138, 3251, 3366

Offset: 0

Author

Bodo Zinser, Feb 10 2005

Keywords

• sign
• easy

Comments

A059100 has a totally different description but is the same sequence (omitting the first two numbers here)

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Daniel Bach ("Dan"), Kissing circles.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A059100, A060790.

Programs

PARI

a(n) = if(n>1, n^2 - 4*n + 6, [-1,2][n+1]) \\ Andrew Howroyd, Feb 25 2018

Formula

a(n) = a(1) + a(2) + a(n-1) + 2*sqrt(a(1)*a(2) + a(1)*a(n-1) + a(2)*a(n-1)) = 1 + a(n-1) + 2*(sqrt(-2 + a(n-1))). (Descartes' curvature-theorem)

From Colin Barker, Jan 07 2013: (Start)

a(n) = n^2 - 4*n + 6 for n > 1.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.

G.f.: -(x^4 + 4*x^3 - 7*x^2 + 5*x - 1)/(x - 1)^3.

(End)