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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

%I

%S 1,2,5,11,25,56,126,283,633,1419,3178,7118,15943,35710,79985,179152,

%T 401270,898777,2013107,4509015,10099422,22620977,50667115,113485664,

%U 254188460,569338636,1275221080,2856276912

%N 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.

%C 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.

%C 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.

%H Kival Ngaokrajang, <a href="/A228765/a228765.pdf">Illustration of initial terms</a>

%F 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

%o (Small Basic)

%o x = 1

%o zeta = (90-45/2)/2

%o c1 = 2*math.Tan(zeta*math.Pi/180)

%o a0 = (1 + Math.SquareRoot(2))/Math.Sin(45*math.pi/180)

%o a1 = a0 - (1 + c1/2)

%o b1 = a1

%o s1 = (a1 + b1 + c1)/2

%o r1 = Math.SquareRoot((s1-a1)*(s1-b1)*(s1-c1)/s1)

%o For n = 0 To 40

%o x[n+1] = x[n] * r1

%o TextWindow.Write(math.Round(1/x[n]) + ", ")

%o EndFor

%o (PARI) a(n)=my(k=7-sqrt(32)+sqrt(80-56*sqrt(2))); round(k^(n-1)) \\ _Charles R Greathouse IV_, Sep 05 2013

%Y Cf. A003499, A156035, A005563.

%K nonn

%O 0,2

%A _Kival Ngaokrajang_, Sep 03 2013

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Last modified May 16 11:54 EDT 2021. Contains 343942 sequences. (Running on oeis4.)