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A331911 Triangle read by rows: Take an equilateral triangle with all diagonals drawn, as in A092867. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+2 and where n is the number of equal parts each side is divided into.

%I A331911 #29 May 13 2020 23:13:37
%S A331911 1,12,0,48,24,3,162,90,0,0,378,306,15,16,0,774,696,84,18,0,0,1470,
%T A331911 1383,219,37,0,0,0,2604,2382,600,78,6,6,0,0,4224,4089,771,177,24,6,0,
%U A331911 0,0,6624,6186,1470,234,42,0,0,0,0,0,9738,9486,2307,498,48,0,0,3,0,1,0,14010,13548,3984,816,144,0,0,0,0,0,0,0
%N A331911 Triangle read by rows: Take an equilateral triangle with all diagonals drawn, as in A092867. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+2 and where n is the number of equal parts each side is divided into.
%H A331911 Hugo Pfoertner, <a href="/A092866/a092866.pdf">Intersections of diagonals in polygons of triangular shape.</a>
%H A331911 Scott R. Shannon, <a href="/A331911/a331911.png">Triangle regions for n = 2</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_1.png">Triangle regions for n = 3</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_13.png">Triangle regions for n = 4</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_3.png">Triangle regions for n = 5</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_4.png">Triangle regions for n = 6</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_5.png">Triangle regions for n = 7</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_6.png">Triangle regions for n = 8</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_7.png">Triangle regions for n = 9</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_8.png">Triangle regions for n = 10</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_10.png">Triangle regions for n = 11</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_9.png">Triangle regions for n = 12</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_11.png">Triangle regions for n = 13</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_12.png">Triangle regions for n = 14</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_14.png">Triangle regions for n = 9, random distance-based coloring</a>.
%H A331911 Scott R. Shannon, <a href="/A331911/a331911_15.png">Triangle regions for n = 12, random distance-based coloring</a>
%e A331911 An equilateral triangle with no other point along its edges, n = 1, contains 1 triangle so the first row is [1]. An equilateral triangle with 1 point dividing its edges, n = 2, contains 12 triangles and no other n-gons, so the second row is [12,0]. An equilateral triangle with 2 points dividing its edges, n = 3, contains 48 triangles, 24 quadrilaterals and 3 pentagons, so the third row is [48,24,3].
%e A331911 Triangle begins:
%e A331911 1
%e A331911 12,0
%e A331911 48,24,3
%e A331911 162,90,0,0
%e A331911 378,306,15,16,0
%e A331911 774,696,84,18,0,0
%e A331911 1470,1383,219,37,0,0,0
%e A331911 2604,2382,600,78,6,6,0,0
%e A331911 4224,4089,771,177,24,6,0,0,0
%e A331911 6624,6186,1470,234,42,0,0,0,0,0
%e A331911 9738,9486,2307,498,48,0,0,3,0,1,0
%e A331911 14010,13548,3984,816,144,0,0,0,0,0,0,0
%e A331911 19248,19224,5007,1102,156,18,0,0,0,0,0,0,0
%e A331911 26208,26142,8634,1668,192,24,0,0,0,0,0,0,0,0
%e A331911 The row sums are A092867.
%Y A331911 Cf. A092867, A092866, A092866, A274586, A331451, A331452, A331907, A331909
%K A331911 nonn,more,tabl
%O A331911 1,2
%A A331911 _Scott R. Shannon_ and _N. J. A. Sloane_, Feb 01 2020