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

Author

Kival Ngaokrajang, Sep 03 2013

Keywords

• nonn

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

• Kival Ngaokrajang, Illustration of initial terms

Crossrefs

Cf. A003499, A156035, A005563.

Programs

PARI

a(n)=my(k=7-sqrt(32)+sqrt(80-56*sqrt(2))); round(k^(n-1)) \\ Charles R Greathouse IV, Sep 05 2013

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