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.

17, 53, 185, 396, 4926, 9086, 20291, 28083, 440835, 579644, 1819320, 3032895, 8305458, 15159436, 29824343, 46104922, 88019569, 89143145, 94929121, 107958869, 227428224, 402409536, 527154160, 2264800904, 2742108435, 3671508788, 7794992942, 52241735601, 202972158380

Offset

1,1

Comments

Any a(i) or a(i)+1 must belong to A072895.

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.

Links

Herbert Kociemba, Table of n, a(n) for n = 1..40

The MacTutor History of Mathematics archive, Theodorus of Cyrene.

The National Museum of American History, Kenneth E. Behring Center, Painting - Square Roots to Sixteen.

Wikipedia, Theodorus of Cyrene.

Example

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.
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°.
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.
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°.

Mathematica

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
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; ];
  If[b<minDist&&b<a, AppendTo[lst, n]; minDist=b; ]; k++]; lst (* Herbert Kociemba, Jul 18 2013 *)

Crossrefs

Cf. A072895.

Sequence in context: A107175 A244270 A244271 * A125637 A320897 A147255

Adjacent sequences: A224266 A224267 A224268 * A224270 A224271 A224272

Keyword

nonn

Author

Robert G. Wilson v, Apr 02 2013

Status

approved