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 A367872 #34 Feb 22 2024 21:00:55
%S A367872 1,4,30,272,2695,28080,302064,3321120,37095201,419276660,4782798020,
%T A367872 54960207120,635339153865,7380876649216,86101923008160,
%U A367872 1007980225327680,11836181297108565,139353762142502100
%N A367872 Number of dissections of a convex (4n+4)-sided polygon into n hexagons and one square (up to equivalence).
%C A367872 This sequence counts dissections of a convex 4n+4-sided polygon into one square and n hexagons, modulo a simple equivalence relation. The equivalence relation is not defined by a group, but by local moves. Consider the octagon formed by a hexagon adjacent to the square. The local move is half-rotation of such octagons.
%C A367872 It seems that a(n) is divisible by n+1.
%F A367872 a(n) = binomial(5*n+2,n)*(n+3)/(4*n+3).
%e A367872 For n=0, there is just one square, so that a(0)=1. For n=1, one can dissect an octagon in 8 ways into a hexagon and a square. In this case, the equivalence relation just relates every such dissection to its half rotated image, so that a(1)=4.
%t A367872 Table[Binomial[5*n + 2, n]*(n + 3)/(4*n + 3), {n, 0, 50}]
%o A367872 (Sage)
%o A367872 def A367872(n):
%o A367872 return binomial(5*n+2, n) * (n+3) / (4*n+3)
%o A367872 (PARI) for(n=0,25, print1(binomial(5*n+2,n)*(n+3)/(4*n+3), ", "))
%Y A367872 Cf. A174687, A185113 (similar), A118970 (related).
%K A367872 nonn
%O A367872 0,2
%A A367872 _F. Chapoton_, Feb 22 2024