cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A296179 Number of points of the inner discrete Theodorus spiral on sheet S_n, n >= 1. First differences of A295339.

Original entry on oeis.org

15, 37, 56, 76, 95, 115, 136, 154, 175, 194, 214, 234, 254, 273, 293, 313, 332, 352, 372, 392, 411, 432, 450, 471, 490, 511, 529, 550, 569, 590, 608, 629, 648, 668, 688, 708, 727, 747, 767
Offset: 1

Views

Author

Wolfdieter Lang, Dec 13 2017

Keywords

Comments

In the complex plane the punctured sheets S_n are given by rho*exp(i*phi_n), with rho > 0 and 2*Pi*(n-1) <= phi_n < 2*Pi*n, for n >= 1.
For the inner discrete Theodorus spiral see the Waldvogel link.
The conjecture stated in A295339 implies that a(n) = A295338(n), for n >= 2.

Crossrefs

Cf. A295339, A295338 (outer spiral), A172164.

Formula

a(n) = b(n) - b(n-1), for n >= 1, with b(n) = A295339(n), and b(0) = 0.
Conjecture: a(n) = A295338(n), for n >= 2 (see a comment above).

A296181 First point of the discrete Theodorus spiral in the fourth quadrant for the n-th revolution, for n >= 1.

Original entry on oeis.org

12, 44, 95, 166, 256, 367, 497, 647, 816, 1006, 1215, 1444, 1692, 1961, 2249, 2557, 2884, 3231, 3598, 3985, 4392, 4818, 5264, 5730, 6215, 6720, 7245, 7790
Offset: 1

Views

Author

Wolfdieter Lang, Jan 05 2018

Keywords

Comments

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

Examples

			a(1) = 12 because phi(11) - 3*Pi/2 is about -0.1869017440 (Maple 10 digits), that is, KIV(11) = -1 + 1 = 0 (not n = 1) but phi(12) - 3*Pi/2 is about +0.1059410277, that is, KIV(12) = 0 + 1 = 1 (on sheet S_1).
a(2) = 44 because  phi(43) - 3*Pi/2 is about 6.270091849, that is KIV(43) = 0 + 1 = 1 (not n = 2) but varphi(44) - 3*Pi/2 is about 6.421424486, that is KIV(44) = 1 + 1 = 2 (on sheet S_2).
		

Crossrefs

Formula

a(n) is the smallest index k for which KIV(k) = n, with KIV(k):= floor((phi(k) - 3*Pi/2)/(2*Pi)) + 1, for k >= 1, where phi(k) is the polar angle of the point z_k = sqrt(n)*exp(i*phi(k)) of the (outer) discrete Theodorus spiral.
Showing 1-2 of 2 results.