A255163 Irrational parts of circle radii in nested circles and hexagons (see comment).

1, 2, -12, 168, -2160, 27936, -361152, 4669056, -60362496, 780378624, -10088893440, 130431264768, -1686241898496, 21800077959168, -281835838291968, 3643630995013632, -47105601999667200, 608990795936169984, -7873156775230046208

Offset: 0

Author

Kival Ngaokrajang, Feb 15 2015

Keywords

• sign

Comments

Inspired by Vitruvian Man, but using hexagons instead of squares, starting with a hexagon whose sides are of length 4 (in some units). The radius of the circle is an integer in the real quadratic number field Q(sqrt(3)), namely R(n) = A(n) + B(n)*sqrt(3) with A(0)=2, A(n) = A255162(n), and B(0) = 1, B(n) = a(n). See illustrations in the links.

Links

• Kival Ngaokrajang, Illustration of initial terms, Vitruvian Man

Crossrefs

Cf. A174968, A170931, A094013, A255162.

Programs

PARI

{a=2;b=1;print1(b,", ");for(n=1,30,c=12*b-6*a;d=4*a-6*b;print1(d,", ");a=c;b=d)}

Formula

Conjectures from Colin Barker, Feb 15 2015: (Start)

a(n) = -12*a(n-1) + 12*a(n-2).

G.f.: -(14*x+1) / (12*x^2-12*x-1).

(End)