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%I #21 Nov 26 2015 05:49:50
%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[0] = 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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