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A224269 Consider the spiral of Theodorus (A072895). This sequence gives the number of k successive triangles which is closer to 360 degrees than any previous k triangles.

%I A224269 #28 Jun 27 2017 15:38:10
%S A224269 17,53,185,396,4926,9086,20291,28083,440835,579644,1819320,3032895,
%T A224269 8305458,15159436,29824343,46104922,88019569,89143145,94929121,
%U A224269 107958869,227428224,402409536,527154160
%N A224269 Consider the spiral of Theodorus (A072895). This sequence gives the number of k successive triangles which is closer to 360 degrees than any previous k triangles.
%C A224269 Any a(i) or a(i)+1 must belong to A072895.
%C A224269 These entries correspond to 1, 2, 4, 6, 22, 30, 45, 53, 211, 242, 429, 554, 917, 1239, 1738, 2161, 2986, 3005, 3101, 3307, 4800, 6385, 7308, ..., turns around the axis. Use the formula in A072895 to check the entries.
%C A224269 Search limit: 10000 turns about the axis.
%H A224269 Herbert Kociemba, <a href="/A224269/b224269.txt">Table of n, a(n) for n = 1..40</a>
%H A224269 The MacTutor History of Mathematics archive, <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Theodorus.html">Theodorus of Cyrene.</a>
%H A224269 The National Museum of American History, Kenneth E. Behring Center, <a href="http://americanhistory.si.edu/collections/search/object/nmah_694656">Painting - Square Roots to Sixteen.</a>
%H A224269 Wikipedia, <a href="http://en.wikipedia.org/wiki/Theodorus_of_Cyrene">Theodorus of Cyrene</a>
%e A224269 a(1) = 17 because the first 16 right triangles result in 351.15042° (8.84958° before the original axis) and the first 17 right triangles result in 364.78344°. 17 right triangles are within 4.78344° of the original axis.
%e A224269 a(2) = 53 because the first 54 right triangles result in 727.48834° and the first 53 right triangles result in 719.73897°.  This is closer to the original axis than 16 and is within 0.2610252°.
%e A224269 a(3) # 109 nor 110 because the first 109 right triangles result in 1079.12463° and the first 110 right triangles result in 1084.57110°. Neither angle is closer to the original axis (1080°) than 53. Therefore the third turn around the center is not close to the original axis than twice around.
%e A224269 a(3) = 185 because the first 186 right triangles result in 1444.08227° (4.08227° after the original axis) and the first 185 right triangles result in 1439.88864°.  This is closer to the original axis than 53 and is within 0.11136°.
%t A224269 lmt = Infinity; lst = {}; k = n = 1; s = 0; While[n < 1001, While[s < 2Pi*n, s = N[s + ArcTan[ 1 / Sqrt@ k], 32]; k++]; a = s - 2Pi*n; b = 2Pi*n - (s - ArcTan[1/Sqrt[k - 1]]); If[Min[a, b] < lmt, lmt = Min[a, b]; If[a < b, AppendTo[lst, {n, k - 1}]; Print[{n, k - 1}], AppendTo[lst, {n, k - 2}]; Print[{n, k - 2}]]]; n++]; Last@ Transpose@ lst
%t A224269 k=minDist=1;lst={};K=-2.1577829966594462209291427868295777235; num[n_]:=Module[{a=-(K/2)+n Pi,b},b=a^2-1/6;If[Floor[b]==Floor[b+1/(144 a^2)],Floor[b],Undefined]]; While[k<40000000,n=num[k];If[!NumberQ[n],Print[k," Stop"];Break[]]; a=2Pi-Mod[K+2 Sqrt[n]+1/(6 Sqrt[n]),2Pi]; b=Mod[K+2 Sqrt[n+1]+1/(6 Sqrt[n+1]),2Pi]; If[a<minDist&&a<b,AppendTo[lst,n-1];minDist=a;];
%t A224269   If[b<minDist&&b<a,AppendTo[lst,n];minDist=b;];k++];lst (* _Herbert Kociemba_, Jul 18 2013 *)
%Y A224269 Cf. A072895.
%K A224269 nonn
%O A224269 1,1
%A A224269 _Robert G. Wilson v_, Apr 02 2013