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A092098 Number of regions that the line segments in A091908(n) cut the equilateral triangle into.

%I A092098 #43 Jul 07 2023 14:37:42
%S A092098 1,6,19,30,61,78,127,150,217,246,331,366,469,510,625,678,817,870,1027,
%T A092098 1080,1261,1326,1519,1566,1801,1878,2107,2190,2437,2520,2791,2886,
%U A092098 3169,3270,3559,3678,3997,4110,4447,4548,4921,5034,5419,5550,5899,6078,6487
%N A092098 Number of regions that the line segments in A091908(n) cut the equilateral triangle into.
%C A092098 Number of chambers in an n-sected triangle. That is, n sectors are extended from each vertex to the opposite edge of the triangle. - _Eric Gottlieb_, Jun 26 2005
%C A092098 How many chambers does the edge n-sected simplex with m vertices have? We have given just the first few terms of the case m = 3. This question is natural in the context of central hyperplane arrangements as it generalizes the braid arrangement. Mike Ackerman, Sul-Young Choi, Peter Coughlin, Japheth Wood and I originally encountered the question in the context of voting theory, where we were exploring ways to tabulate votes when voters' preferences are partially ordered. Unfortunately, it turns out that the chambers of the 3-sected simplex with n vertices are not in correspondence with the set of posets on n letters as the chain with three elements and a fourth incomparable element illustrates. - _Eric Gottlieb_, Jun 26 2005
%C A092098 "Equilateral" is not needed: the sequence counts regions correctly for any triangle with n-sected sides. Ceva's Theorem is used to deduct vanishing regions from the naive count. The first deduction is at n=15 for n odd and n=20 for n even. - _Len Smiley_ and Brian Wick (mathclub(AT)math.uaa.alaska.edu), Jul 04 2005
%H A092098 Hugo Pfoertner, <a href="/A092098/b092098.txt">Table of n, a(n) for n = 1..1000</a>
%H A092098 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a091908.pdf">Visualization of diagonal intersections in an equilateral triangle.</a>
%H A092098 Hugo Pfoertner, <a href="/A092098/a092098.pdf">Visualization of diagonal intersections in an equilateral triangle.</a> [Local copy]
%H A092098 Scott R. Shannon, <a href="/A092098/a092098.jpg">Image for n = 6</a>.
%H A092098 Scott R. Shannon, <a href="/A092098/a092098_1.jpg">Image for n = 7</a>.
%H A092098 Scott R. Shannon, <a href="/A092098/a092098_2.jpg">Image for n = 12</a>.
%H A092098 Scott R. Shannon, <a href="/A092098/a092098_3.jpg">Image for n = 13</a>.
%H A092098 Scott R. Shannon, <a href="/A092098/a092098_4.jpg">Image for n = 100</a>.
%H A092098 Scott R. Shannon, <a href="/A092098/a092098_5.jpg">Image for n = 101</a>.
%H A092098 <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a>
%F A092098 Note that 3 divides a(2k) and a(2k+1)-1. - _T. D. Noe_, Jun 29 2005
%e A092098 E.g. the number of chambers in the bisected triangle is six, the number of permutations on 3 letters. The number of chambers in the trisected triangle is equal to 19, the number of posets on 3 elements. - _Eric Gottlieb_, Jun 26 2005
%e A092098 a(2)=6: The 3 line segments cut the equilateral triangle into 6 triangles.
%e A092098 a(3)=19: The 3*2 line segments form 12 triangles, 3 quadrilaterals, 3 pentagons and 1 central non-regular hexagon. See pictures at Pfoertner link.
%p A092098 regions:=proc(n::nonnegint)
%p A092098    local j,k,l,a;
%p A092098    a:=0;
%p A092098    if (n mod 2<>0) then
%p A092098       a:=3*n^2-3*n+1
%p A092098    else
%p A092098       a:=3*n^2-6*n+6
%p A092098    fi;
%p A092098    for l from 1 to floor(n/2)-1 do
%p A092098       for k from 1 to floor(n/2)-1 do
%p A092098          for j from 1 to floor(n/2)-1 do
%p A092098             if((n-k)*l*j=k*(n-l)*(n-j)) then
%p A092098                a:=a-6
%p A092098             fi
%p A092098          od
%p A092098       od
%p A092098    od;
%p A092098    return a
%p A092098 end proc;
%p A092098 seq(regions(i),i=1..100);  # _Len Smiley_ and Brian Wick, Jun 30 2005
%t A092098 regions[n_]:=
%t A092098 If[Mod[n,2] == 0, 3n^2-6n+6, 3n^2-3n+1]-
%t A092098    6*Count[
%t A092098   Flatten@
%t A092098    Table[
%t A092098     Abs[(n-k)l*j - k(n-l)(n-j)],
%t A092098     {j,1,Floor[n/2]-1},
%t A092098     {k,1,Floor[n/2]-1},
%t A092098     {l,1,Floor[n/2]-1}],
%t A092098   0] (* _Ethan Beihl_, Oct 13 2016 *)
%o A092098 (PARI) for(n=1,100,regions=0;if(n%2!=0,regions=3*n^2-3*n+1,regions=3*n^2-6*n+6);for(l=1,floor(n/2)-1,for(k=1,floor(n/2)-1,for(j=1,floor(n/2)-1,if((n-k)*l*j==k*(n-l)*(n-j),regions-=6))));print1(regions,",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 27 2006
%Y A092098 Cf. A091908 (number of intersections), A091910 (radial locations of intersection points), A006533.
%K A092098 nonn
%O A092098 1,2
%A A092098 _Hugo Pfoertner_, Feb 19 2004
%E A092098 More terms from _T. D. Noe_, Jun 29 2005
%E A092098 Further terms from Brian Wick (mathclub(AT)math.uaa.alaska.edu), Jun 30 2005
%E A092098 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 27 2006