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.
%I A293260 #18 Aug 05 2025 00:13:42
%S A293260 0,0,0,0,0,0,0,1,0,2,0,12,0,9,0,14,0,75,0,26,0,35,0,110,0,54,0,57,0,
%T A293260 436
%N A293260 Number of adventitious quadrangles (convex, noncyclic, not kite) such that Pi/n is the largest number that divides all the angles.
%C A293260 "All the angles" in the title means any angle formed by 3 vertices. There are 8 nonoverlapping angles in total.
%C A293260 Consider convex quadrilateral ABCD. Let a,b,c,d,e,f,g,h be the angles ABD,DBC,BCA,ACD,CDB,BDA,DAC,CAB, respectively. A quadrangle is adventitious if all these angles are rational multiples of Pi.
%C A293260 Cyclic quadrilaterals have properties a=d, b=g, c=f, e=h, thus making the adventitious case trivial.
%C A293260 Kites have properties a=b, c=h, d=g, e=f, thus making the adventitious case trivial.
%C A293260 Some properties:
%C A293260 1. b+c = f+g := x, d+e = h+a := y, x+y = Pi.
%C A293260 2. sin(a)sin(c)sin(e)sin(g) = sin(b)sin(d)sin(f)sin(h).
%C A293260 3. In an adventitious quadrangle, swapping angles (b,c) with (f,g) or (a,h) with (d,e) gives another adventitious quadrangle.
%C A293260 From empirical observation, it seems that no adventitious quadrangles exist for odd numbers n. For example, take n=9: 180 degrees/9 = 20 degrees, and forming a quadrangle in which all angles are multiples of 20 degrees is impossible (proven by brute force). It seems to hold for all odd numbers n.
%C A293260 Perhaps the most famous case is Langley's problem (where n=18).
%H A293260 Kevin S. Brown's Mathpages, <a href="http://www.mathpages.com/home/kmath734/kmath734.htm">Adventitious Solutions</a>
%H A293260 Wikipedia, <a href="https://en.wikipedia.org/wiki/Langley%E2%80%99s_Adventitious_Angles">Langley’s Adventitious Angles</a>
%e A293260 a(8) = 1 because there is one quadrangle where all angles are divisible by 180/8 = 22.5 degrees.
%e A293260 a=90, b=45, c=22.5, d=45, e=67.5, f=45, g=22.5, h=22.5.
%e A293260 a(10) = 2 (180/10 = 18):
%e A293260 72 54 18 36 72 36 36 36
%e A293260 108 36 18 54 72 36 18 18
%e A293260 a(12) = 12 (180/12 = 18):
%e A293260 75 30 45 45 60 60 15 30
%e A293260 75 60 15 45 60 30 45 30
%e A293260 90 30 30 45 75 45 15 30
%e A293260 90 45 15 45 75 30 30 30
%e A293260 90 45 30 45 60 60 15 15
%e A293260 90 45 30 75 30 45 30 15
%e A293260 90 60 15 45 60 45 30 15
%e A293260 105 30 15 30 105 30 15 30
%e A293260 105 30 30 75 45 45 15 15
%e A293260 105 45 15 30 90 45 15 15
%e A293260 105 45 15 75 45 30 30 15
%e A293260 120 30 15 60 75 30 15 15
%t A293260 Remove[f];
%t A293260 f[n_Integer] := Do[
%t A293260 If[A == B < n/2 - C, Continue[]];(* if A == B then C >= H *)
%t A293260 If[A == B == n/2 - C || C == D == n/2 - B, Continue[]];(* remove kite *)
%t A293260 F = n/\[Pi] ArcTan[(Sin[d] Sin[a + b])/(Sin[a] Sin[c] Sin[e]) -
%t A293260 Cot[e], 1] /. Thread[{a, b, c, d, e} -> \[Pi]/n {A, B, C, D, E}];
%t A293260 F = Round[F, 10^-6];
%t A293260 If[A < F, Continue[]];
%t A293260 If[GCD[A, B, C, D, E, F] != 1, Continue[]];
%t A293260 If[A == E && B < F, Continue[]];(* if A == E then B >= F *)
%t A293260 If[A == F && B < E, Continue[]];(* if A == F then B >= E *)
%t A293260 {A, B, C, D, E, F, B + C - F, D + E - A} // Sow;
%t A293260 , {A, n/4 // Ceiling, n - 3}
%t A293260 , {B, Max[1, n - 3 A + 2], Min[A, n - A - 2]}(* B <= A and C < A and H < A *)
%t A293260 , {C, Max[1, n - 2 A - B + 1], Min[A - 1, n - A - B - 1]}(* C < A and H < A *)
%t A293260 , {D, n - A - B - C, A - 1}(* D < A and E <= A *)
%t A293260 , {E, {n - B - C - D}}
%t A293260 ] // Reap // Last // If[# == {}, {}, # // Last] &;
%t A293260 Table[f[n] // Length, {n, 30}]
%t A293260 (* 180/n f[n] /. n -> 18 // TableForm *)
%K A293260 nonn,more
%O A293260 1,10
%A A293260 _Albert Lau_, Oct 04 2017