A228765 The curvature of circles (rounded to nearest integer), successively inscribed toward the 45-degree angle of a 45-45-90 triangle, starting with a unit circle.

1, 2, 5, 11, 25, 56, 126, 283, 633, 1419, 3178, 7118, 15943, 35710, 79985, 179152, 401270, 898777, 2013107, 4509015, 10099422, 22620977, 50667115, 113485664, 254188460, 569338636, 1275221080, 2856276912

Offset

0,2

Comments

The curvature expansion factors are 2.239828809...(1/0.44636269217...) and 5.828427125...(1/0.17157287525...) or 1 / (3 - 2*sqrt(2)) for circles successively inscribed toward the 45- and 90-degree angles respectively. The ratio 1 / (3 - 2*sqrt(2)) is also 3 + 2*sqrt(2) or A156035 as commented by Michel Marcus. This is also (n+1) + sqrt(A005563(n)) or 1 / ((n+1) - sqrt(A005563(n))), for n = 2.

The curvature of circles (rounded to nearest integer) successively inscribed toward the 90-degree angle is A003499. (except the first term). See illustration in links.

Links

Table of n, a(n) for n=0..27.

Kival Ngaokrajang, Illustration of initial terms

Formula

a(n+1) = round(k^n), with k = 7 - 4 sqrt(2) + 2 sqrt(20 - 14 sqrt(2)) = 2.23982.... - Charles R Greathouse IV, Sep 05 2013

Prog

(Small Basic)
x[0] = 1
zeta = (90-45/2)/2
c1 = 2*math.Tan(zeta*math.Pi/180)
a0 = (1 + Math.SquareRoot(2))/Math.Sin(45*math.pi/180)
a1 = a0 - (1 + c1/2)
b1 = a1
s1 = (a1 + b1 + c1)/2
r1 = Math.SquareRoot((s1-a1)*(s1-b1)*(s1-c1)/s1)
For n = 0 To 40
  x[n+1] = x[n] * r1
  TextWindow.Write(math.Round(1/x[n]) + ", ")
EndFor
(PARI) a(n)=my(k=7-sqrt(32)+sqrt(80-56*sqrt(2))); round(k^(n-1)) \\ Charles R Greathouse IV, Sep 05 2013

Crossrefs

Cf. A003499, A156035, A005563.

Sequence in context: A215091 A017919 A017920 * A006054 A106805 A356846

Adjacent sequences: A228762 A228763 A228764 * A228766 A228767 A228768

Keyword

nonn

Author

Kival Ngaokrajang, Sep 03 2013

Status

approved