This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A351103 #72 Feb 19 2022 04:53:06
%S A351103 0,0,0,3,7,12,3,10,22,3,13,35,3,16,51,3,19,70,3,22,92,3,25,117,3,28,
%T A351103 145,3,31,176,3,34,210,3,37,247,3,40,287,3,43,330,3,46,376,3,49,425,3,
%U A351103 52,477,3,55,532,3,58,590,3,61,651,3,64,715,3,67,782,3,70,852,3,73,925
%N A351103 a(n) is the total number of polygons left over with maximum number of sides when partitioning the set of vertices of a convex n-gon into nonintersecting polygons.
%C A351103 Alternatively, total number of regions left over when partitioning the set of vertices of a convex n-gon into nonintersecting polygons, each containing adjacent vertices of the n-gon.
%F A351103 For k >= 3, a(3*k) = 3, a(3*k+1) = 3*k+1, a(3*k+2) = 3 + a(3*k-2) + a(3*k-1), with a(3)=a(4)=a(5)=0 and a(6) = 3, a(7) = 7, a(8) = 12.
%e A351103 n = 18 is an 18-gon, which has 6 triangles each containing adjacent vertices of the 18-gon and the leftover region in each case is a 12-gon. Since there are only 3 orientations of partitioning, the total number of leftover regions is 3.
%e A351103 n = 19 is a 19-gon, which has 5 triangles and 1 quadrilateral each containing adjacent vertices of the 19-gon and the leftover regions in each case is a 12-gon. Since there are 19 orientations of partitioning, the total number of leftover regions is 19.
%e A351103 n = 20 is a 20-gon, which has 5 triangles with one pentagon or 4 triangles with 2 quadrilaterals. In the first case the number of leftover regions is 20 because it has 20 orientations, in the second case the number of leftover regions is 20 + 20 + 10 = 50 because it has 3 different permutations of 3,3,3,3,4,4 with 20 orientations, 3,3,3,4,3,4 with 20 orientations, and 3,3,4,3,3,4 with 10 orientations. Therefore the total is 70.
%t A351103 Nest[Append[#1, Switch[Mod[#2, 3], 0, 3, 1, #2, 2, 3 + Total@ #1[[3 Quotient[#2, 3] - 4 ;; 3 Quotient[#2, 3] - 3]]]] & @@ {#, Length[#] + 3} &, ConstantArray[0, 3]~Join~{3, 7, 12}, 66] (* _Michael De Vlieger_, Feb 04 2022 *)
%o A351103 (PARI) a(n) = if (n==6, 3, if (n==7, 7, if (n==8, 12, my(x=n%3); if (x==0, 3, if (x==1, n, 3 + a(n-4) + a(n-3)))))); \\ _Michel Marcus_, Feb 01 2022
%Y A351103 Total number of left out regions is A347862.
%K A351103 nonn
%O A351103 3,4
%A A351103 _Janaka Rodrigo_, Jan 31 2022