A277402 "3-Portolan numbers": number of regions formed by n-secting the angles of an equilateral triangle.

1, 6, 19, 30, 61, 78, 127, 150, 217, 234, 331, 366, 469, 510, 631, 678, 817, 870, 1027, 1074, 1261, 1326, 1519, 1590, 1801, 1878, 2107, 2190, 2437, 2514, 2791, 2886, 3169, 3270, 3571, 3678, 3997, 4110, 4447, 4554, 4921, 5046, 5419, 5550, 5941, 6078, 6487, 6630, 7057, 7194

Offset

1,2

Comments

I like the name "portolan numbers": cf. the rhumbline designs on medieval maps, constructed in a similar way.

The regions can be counted using an adaptation of Smiley and Wick's method in A092098: count regions assuming there are no redundant intersections, then subtract the number of regions that Ceva's Theorem says must vanish.

Off-diagonal redundant intersections occur for triples of integers 1 <= i, j, k < floor(n/2)-1 such that M(i)*M(j) = M(k), where M(x) is the ratio (sin(Pi(n-x)/(3n)))/(sin(Pi*x/(3n))). In the case 10|n, this corresponds to the charming identity (sin(7*Pi/30)*sin(8*Pi/30))/(sin(3*Pi/30)*sin(2*Pi/30)) = sin(9*Pi/30)/sin(1*Pi/30).

Differs from A092098 (which counts regions when *sides*, not angles, are n-sected) for the first time at the tenth term.

The above equation has solutions if and only if 10|n. This can be shown by rewriting the equation in exponential form, and using facts about vanishing sums of roots of unity to narrow the possibilities for n. (See Conway and Jones, 1976.) This is computationally feasible because A273096(6) = 1. - Ethan Beihl, Nov 26 2016

Links

Lars Blomberg, Table of n, a(n) for n = 1..500

Lars Blomberg, Coloured illustration for n=3

Lars Blomberg, Coloured illustration for n=4

Lars Blomberg, Coloured illustration for n=19

Lars Blomberg, Coloured illustration for n=20

J. H. Conway and A. J. Jones, Trigonometric diophantine equations (On vanishing sums of roots of unity), Acta Arithmetica 30(3), 229-240 (1976).

Wikipedia, Rhumbline network

Formula

Empirical g.f.: x*(1 + 5*x + 12*x^2 + 6*x^3 + 18*x^4 + 6*x^5 + 18*x^6 + 6*x^7 + 18*x^8 - 6*x^9 + 29*x^10 + 13*x^11 - 6*x^12) / ((1-x)^3*(1+x)^2*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Oct 14 2016

Empirically for 12 < n <= 500: a(n) = a(n-2) + a(n-10) - a(n-12) + 120. - Lars Blomberg, Jun 08 2020

Empirical: a(2*k + 1) = 6*k*(2*k + 1) + 1, for k >= 0. - Ivan N. Ianakiev, Jun 27 2020

Empirical: 10*a(n) = 30*n^2 -45*n +23 +13*(-1)^n -15*(-1)^n*n - 24*b(n) where b(n) is the 10-periodic sequence 4, 0, -1, 0, -1, 0, -1, 0, -1, 0, 4, 0 .... of offset 0. - R. J. Mathar, Jul 05 2020

Example

For n=3, a(n) gives the 19 regions formed by the intersection of 3*2 lines: 3 pentagons, 3 quadrilaterals, 12 triangles, and 1 big central hexagon.

Mathematica

regions[n_]:=
If[Mod[n, 2]==0, 3n^2-6n+6, 3n^2-3n+1]-
  6*Length@
    Select[
     Flatten@
      With[
       {b=N@
          Table[
             1/2 - (Sqrt[3]/2)Tan[(60Degree/n)(n/2-i)],
             {i, 1, Floor[n/2]- 1}
             ]},
       Table[
        Abs[(1-b[[k]])b[[l]]b[[j]] - b[[k]](1-b[[l]])(1-b[[j]])],
        {j, 1, Floor[n/2]-1},
        {k, 1, Floor[n/2]-1},
        {l, 1, Floor[n/2]-1}]
       ],
     Chop@#==0&]

Crossrefs

Cf. A092098, A335411 (vertices), A335412 (edges), A335413 (ngons).

Sequence in context: A210288 A038125 A319968 * A092098 A186113 A162332

Adjacent sequences: A277399 A277400 A277401 * A277403 A277404 A277405

Keyword

nonn

Author

Ethan Beihl, Oct 13 2016

Status

approved