login
Irregular table read by rows: Take a nonagon with all diagonals drawn, as in A332421. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.
5

%I #26 Jan 01 2024 22:47:11

%S 90,36,18,9,0,0,1,1332,918,414,90,6525,6453,2529,1071,171,90,10,9,

%T 22248,18882,10368,2988,486,108,18,54558,50985,24750,9387,2034,531,36,

%U 27,9,0,0,0,0,0,0,1,113958,107676,54558,17820,3672,612,36,18

%N Irregular table read by rows: Take a nonagon with all diagonals drawn, as in A332421. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

%C See the links in A332421 for images of the nonagons.

%H Lars Blomberg, <a href="/A332427/b332427.txt">Table of n, a(n) for n = 1..241</a> (the first 23 rows)

%e A nonagon with no other points along its edges, n = 1, contains 90 triangles, 36 quadrilaterals, 18 pentagons, 9 hexagons, 1 nonagon and no other n-gons, so the first row is [90,36,18,9,0,0,1]. A nonagon with 1 point dividing its edges, n = 2, contains 1332 triangles, 918 quadrilaterals, 414 pentagons, 90 hexagons and no other n-gons, so the second row is [1332,918,414,90].

%e Table begins:

%e 90,36,18,9,0,0,1;

%e 1332,918,414,90;

%e 6525,6453,2529,1071,171,90,10,9;

%e 22248,18882,10368,2988,486,108,18;

%e 54558,50985,24750,9387,2034,531,36,27,9,0,0,0,0,0,0,1;

%e 113958,107676,54558,17820,3672,612,36,18;

%e 210591,208089,105417,34407,7560,1737,307,45,0,9;

%e The row sums are A332421.

%Y Cf. A332421 (regions), A332428 (vertices), A332429 (edges), A007678, A092867, A331452, A331929.

%K nonn,tabf

%O 1,1

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 09 2020

%E a(36) and beyond from _Lars Blomberg_, May 16 2020