A331907 - OEIS

%I #15 May 06 2020 04:58:42

%S 40,0,0,590,420,80,10,2890,3030,1130,230,50,9540,10530,4290,980,190,

%T 10,22730,28390,10960,3200,550,80,20,47610,57450,23270,6530,1160,160,

%U 20,0,90080,109160,47430,13430,2460,410,40,0,0,154840,193480,82330,22410,4620

%N Triangle read by rows: Take a pentagram with all diagonals drawn, as in A331906. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+2.

%C See the links in A331906 for images of the pentagrams.

%H Lars Blomberg, <a href="/A331907/b331907.txt">Table of n, a(n) for n = 1..250</a> (the first 20 rows)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pentagram.html">Pentagram</a>.

%e A pentagram with no other points along its edges, n = 1, contains 40 triangles and no other n-gons, so the first row is [40,0,0]. A pentagram with 1 point dividing its edges, n = 2, contains 590 triangles, 420 quadrilaterals, 80 pentagons and 10 hexagons, so the second row is [590,420,80,10].

%e Triangle begins:

%e 40,0,0

%e 590, 420, 80, 10

%e 2890, 3030, 1130, 230, 50

%e 9540, 10530, 4290, 980, 190, 10

%e 22730, 28390, 10960, 3200, 550, 80, 20

%e 47610, 57450, 23270, 6530, 1160, 160, 20, 0

%e The row sums are A331906.

%Y Cf. A331906 (regions), A333117 (vertices), A333118 (edges), A007678, A092867, A331452.

%K nonn,tabf

%O 1,1

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 31 2020

%E a(34) and beyond from _Lars Blomberg_, May 06 2020