A255163 -id:A255163 - cp's OEIS frontend

A255162 Rational part of circle radii in nested circles and hexagons (see comment).

2, 0, 24, -288, 3744, -48384, 625536, -8087040, 104550912, -1351655424, 17474476032, -225913577472, 2920656642048, -37758842634240, 488153991315456, -6310954007396352, 81589295984541696, -1054802999903256576, 13636707550653579264

Offset: 0

Kival Ngaokrajang, Feb 15 2015

sign

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) = a(n), and B(0) = 1, B(n) = A255163(n). See illustrations in the links.

Kival Ngaokrajang, Illustration of initial terms, Vitruvian Man

Cf. A174968, A170931, A094013, A255163.

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

Conjectures from Colin Barker, Feb 15 2015: (Start)
a(n) = -12*a(n-1) + 12*a(n-2).
G.f.: -2*(12*x+1) / (12*x^2 - 12*x - 1).
(End)

A255405 a(n) = floor((2/sqrt(Pi))^n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 23, 26, 29, 33, 37, 42, 47, 53, 60, 68, 77, 87, 98, 111, 125, 141, 159, 180, 203, 229, 258, 292, 329, 371, 419, 473, 534, 602, 680, 767, 865, 977, 1102, 1244, 1403, 1584, 1787, 2016, 2275, 2567

Offset: 0

Kival Ngaokrajang, Feb 22 2015

nonn

Inspired by squaring the circle and Vitruvian Man, but starting with a unit circle and a square whose sides are of length sqrt(Pi), A002161. a(n) is the curvature (rounded down) of the n-th circle. See illustrations in the links.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Kival Ngaokrajang, Illustration of initial terms.
Wikipedia, Squaring the circle.

Cf. A002161, A255162, A255163.

Table[Floor[(2/Sqrt[Pi])^n], {n,0,50}] (* G. C. Greubel, Jan 09 2017 *)

{for(n=1,100,a=floor(2^n/sqrt(Pi)^n);print1(a,", "))}

a(n) = floor((2/sqrt(Pi))^n).

cp's OEIS Frontend

A255162 Rational part of circle radii in nested circles and hexagons (see comment).

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