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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
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 (list; graph; refs; listen; history; text; internal format)
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 A094981

Adjacent sequences:  A228762 A228763 A228764 * A228766 A228767 A228768

KEYWORD

nonn

AUTHOR

Kival Ngaokrajang, Sep 03 2013

STATUS

approved

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Last modified June 24 05:58 EDT 2021. Contains 345416 sequences. (Running on oeis4.)