cp's OEIS Frontend

Here the points of the inner discrete Theodurus spiral in the complex plane are zhat(k) = rho(k)*exp(i*phihat(k)) with rho(k) = sqrt(k) and phihat(k) starts with phihat(1) = Pi/2 and is not restricted to be <= 2*Pi, it is phihat(k) = Sum_{j=0..k-1} (2*alpha(j+1) - alpha(j)) with alpha(j) = arctan(1/sqrt(j)), for k >= 1. The formula is phihat(k) = phi(k) + alpha(k), with the recurrence for the arguments of the outer spiral phi(k) = phi(k-1) + alpha(k-1), k >= 2, with phi(1) = 0.

If one considers punctured sheets S_n = rho*exp(i*phi_n), with rho > 0 and 2*Pi*(n-1) <= phi_n < 2*Pi*n, for n >= 1, then on sheet S_n there are a(n) - a(n-1) = A296179(n) points zhat, where a(0) = 0.

An analytic continuation of Davis's interpolation of the outer spiral is given in the Waldvogel link (see Figure 2 there). The point zhat(k) (called G_k on Figure 1 there) on the inner spiral is obtained from mirroring the point z(k) (called F_k there) of the outer spiral on the hypotenuse O,z(k+1), for k >= 1. In the present case the arguments phihat(k) of zhat(k) are taken positive.

Conjecture: a(n) = A072895(n) - 2, n >= 1. This follows from the conjecture that the sequences K := {floor(phi(k)/(2*Pi)}{k >= 1} with phi given above, and Khat:= {floor(phihat(k)/(2*Pi)}{k >= 1} with phihat given above satisfy Khat(k-2) = K(k), for k >= 3. Note that phihat(k-2) - phi(k) = alpha(k-2) - alpha(k-1) =: delta(k) = arctan((sqrt(k-1) - sqrt(k-2))/(1 + sqrt((k-1)*(k-2)))) > 0, for k >= 3. Therefore the conjecture is that delta(k) < 2*Pi*(1 - frac(phi(k)/(2*Pi))), for k >= 3, or, equivalently, phihat(k-2) < 2*Pi*(K(k) + 1), for k >= 3.

This sequence is used in a conjecture on points z_k of the discrete (outer) Theodorus spiral living on quadrant IV of the complex plane of sheet S_n, where S_n := {r*exp(i*phi), r > 0, 2*Pi*(n-1) <= phi < 2*Pi*n}. This corresponds to the n-th revolution, for n >= 1.

This conjecture is 2*Pi - varphi(A072895(n)) > arctan(a(n)), n >= 1, with varphi(k) = phi(k) - 2*Pi*floor(phi(k)/(2*Pi)) where z_k = sqrt(k)*exp(i*phi(k)).

This conjecture implies a conjecture relating points of the discrete inner spiral to those of the outer ones, namely Khat(k-2) := floor(phihat(k-2)/(2*Pi)) = K(k) =: floor(phi(k)/(2*Pi)) for k >= 3, where zhat_k = sqrt(k)*exp(i*phihat(k)) is a point of the discrete inner Theodorus spiral, given in terms of z_k by zhat(k) = ((k-1 + 2*sqrt(k)*i )/(k+1))*z_k. This implies phihat(k) = phi(k) + arctan((sqrt(k-1) - sqrt(k-2))/(1 + sqrt((k-1)*(k-2)))). The implied conjecture Khat(k-2) = K(k), k >= 3, for the other three quadrants of each sheet S_n can be proved. For the inner spiral see the Waldvogel link.

If the implied conjecture is true then A295339(n) = A072895(n) - 2, for n >= 1, hence A296179(n) = A295338(n), for n >= 2.

For the conjecture and the proof for the first three quadrants for each sheet S_n see the W. Lang link. - Wolfdieter Lang, Jan 24 2018