A295339 Least k for the inner Theodorus spiral to complete n revolutions.

15, 52, 108, 184, 279, 394, 530, 684, 859, 1053, 1267, 1501, 1755, 2028, 2321, 2634, 2966, 3318, 3690, 4082, 4493, 4925, 5375, 5846, 6336, 6847, 7376, 7926, 8495, 9085, 9693, 10322, 10970, 11638, 12326, 13034, 13761, 14508, 15275

Offset: 1

Author

Wolfdieter Lang, Dec 13 2017

Keywords

• nonn

Comments

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.

References

• P. J. Davis, Spirals from Theodorus to Chaos, A K Peters, Wellesley, MA, 1993.

Links

• Joerg Waldvogel, Analytic Continuation of the Theodorus Spiral.

Crossrefs

Cf. A072895 (outer spiral), A296179.

Formula

a(n) = -1 + first position of n in the sequence

Khat:= {floor(phihat(k)/(2*Pi))}_{ k>= 1}, with phihat given in a comment above in terms of phi.

Conjecture: a(n) = A072895(n) - 2, n >= 1 (see the comment above).