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This sequence also gives the curvature of touching circles inscribed in a special way in the smaller segment of a circle of radius 5/4 cut by a chord of length 2.

Consider a circle C of radius 5/4 (in some length units) with a chord of length 2. This has been chosen so that the larger sagitta also has length 2. The smaller sagitta has length 1/2. The input, besides the circle C, is the circle C_0 with radius R_0 = 1/4, touching the chord and circle C. The following sequence of circles C_n with radii R_n, n >= 1, is obtained from the conditions that C_n touches (i) the circle C, (ii) the chord and (iii) the circle C_(n-1). The curvature of the n-th circle, C_n = 1/R_n, n >= 0, is conjectured to be a(n). See an illustration given in the link. As found by Wolfdieter Lang (see part II of the proof given by W. Lang in the link), this circle problem is related to the nonnegative solutions of the Pell equation X^2 - 5*Y^2 = 4: a(n) = 2 + X(n) = 2 + A005248(n). For the larger segment below the chord (with sagitta length 2) the sequence would be A115032, see W. Lang's proof given in part I of the link.

If the circle radius and the sagitta length were both equal to 1, the curvature sequence would be A099938.

Essentially a duplicate of A092387. - R. J. Mathar, Jul 07 2023

This sequence is motivated by Kival Ngaokrajang's touching circle problem considered in A240926 and A115032.

a(n), together with b(n) = A246639(n) appears as the curvature c(n) = a(n) + (4*b(n)/5)*phi (phi = (1+sqrt(5))/2, golden section) of the circle which touches i) a circle of radius 5/4 (in some length units) divided by a chord of length 2 into two unequal parts, and ii) the two touching circles in the smaller part which have curvatures A240926(n) and A240926(n+1), both also touching the circle with radius 5/4. See the illustration of Kival Ngaokrajang's link given in A240926, where the first circles in the smaller (upper) part are shown. The present circles will lie in the region between the large circle and two of these circles in the upper part.

Descartes' theorem on touching circles (see the links) is applied here as c(n) = -4/5 + A(n) + A(n+1) + 2*sqrt((-4/5 )*(A(n) + A(n+1)) + A(n)*A(n+1)), with A(n) = A240926(n), n >= 0.

For the proof for the first formula for a(n) given below use the formula for the curvature A240926(n) = 2 + 2*S(n, 3) - 3* S(n-1, 3) (see the W. Lang link in A240926, part II) in c(n) and compare with a(n) from c(n) = a(n) + (4*b(n)/5)*phi. This is done by using standard S-polynomial identities like the three term recurrence and the Cassini-Simson type identity

S(n, x)*S(n-2, x) = -1 + S(n-1, x)^2 (here for x=3). This implies S(n, 3)*S(n-1, 3) = (-1 + S(n, 3)^2 + S(n-1, 3)^2)/3. See also the mentioned link, part III a).

a(n) appears also in the curvature for the touching circles and chord problem in the smaller part of a circle with radius 5/4 dissected by a chord of length 2, together with A246640, where details are given.