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.

A352741 Numbers on the Spiral of Theodorus that come closer to the x-axis than their neighbors; a(0) = 0 and a(1) = 1.

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%I A352741 #28 May 13 2022 11:42:26
%S A352741 0,1,7,18,33,54,80,110,146,186,231,282,337,397,462,532,607,687,772,
%T A352741 861,956,1056,1160,1270,1384,1504,1628,1757,1891,2030,2174,2323,2477,
%U A352741 2636,2800,2969,3142,3321,3504,3693,3886,4084,4288,4496,4709,4927,5150,5378,5611,5849,6091,6339
%N A352741 Numbers on the Spiral of Theodorus that come closer to the x-axis than their neighbors; a(0) = 0 and a(1) = 1.
%C A352741 The Spiral of Theodorus (a.k.a. Pythagorean spiral or Pythagoras's snail) results from constructing the square roots sqrt(2), sqrt(3), sqrt(4), ... as hypotenuse of the right triangle having the previous one as longer leg, and the shorter leg equal to 1, starting with the segment [0, 1] for sqrt(1). In the complex plane, this construction corresponds to a sequence z(n+1) = z(n) + i*z(n)/|z(n)|, starting with z(1) = 1 (and z(0) = 0 by convention).
%C A352741 This sequence lists those n for which the endpoint is closer to the x-axis than the preceding and next one (plus the two initial points 0 and 1 which are the only ones lying directly on the x-axis). In terms of the complex sequence z(n), this means the indices n such that abs(Im(z(n)) <=  abs(Im(z(n +- 1))).
%C A352741 The two initial terms correspond to the points (0, 0) and (1, 0), which can be regarded as the start of the Pythagorean spiral, and are the only points exactly on the x-axis.
%C A352741 There is a smooth complex-valued function z: [0,oo) -> C, t |-> z(t), which interpolates the spiral in non-integer values. In terms of this function, a(n) = round(t(n)) where t(n) is the n-th zero of the imaginary part of z (if indexing starts with 0 for t(0) = 0, then t(1) = 1, and t(2) ~ 6.8 where z(6.8) = -1). (This function has a natural extension to the whole of R including also the negative real line, but we don't consider negative arguments here.)
%H A352741 Christian Mercat, <a href="https://www.geogebra.org/m/fGnDEWaR">Escargot de Pythagore: Spirale des racines carrées d'entiers</a>, GeoGebra.org, Dec. 12, 2015
%H A352741 Wikipedia, <a href="https://en.wikipedia.org/wiki/Spiral_of_Theodorus">Spiral of Theodorus</a>, as of March 14, 2022
%F A352741 a(2n+1) = A072895(n) or A072895(n)+1, for all n > 0.
%o A352741 (PARI) z=!t=y=0; [n | n<-[0..10^4], ![t < t=y, y > y=abs(imag(z))]*z*=1+I/abs(z)]
%Y A352741 Cf. A072895, A137515, A105459 (Hlawka's "snail constant").
%K A352741 nonn
%O A352741 0,3
%A A352741 _M. F. Hasler_, Mar 31 2022