cp's OEIS Frontend

A320431 The number of tiles inside a regular n-gon created by lines that run from each of the vertices of the n edges orthogonal to these edges.

%I A320431 #25 Sep 15 2021 06:22:55
%S A320431 1,1,31,13,71,25,127,41,199,61,287,85,391,113,511,145,647,181,799,221,
%T A320431 967,265,1151,313,1351,365,1567,421,1799,481,2047,545,2311,613,2591,
%U A320431 685,2887,761,3199,841,3527,925,3871,1013,4231,1105,4607,1201,4999,1301,5407,1405,5831,1513,6271,1625,6727,1741
%N A320431 The number of tiles inside a regular n-gon created by lines that run from each of the vertices of the n edges orthogonal to these edges.
%C A320431 Sequence proposed by Thomas Young: draw the regular n-gon and construct 2*n lines that run from both ends of the n edges perpendicular into the n-gon until they hit an opposite edge. (For n even the lines actually hit another vertex, so there are only n additional lines). a(n) is the number of non-overlapping tiles inside the n-gon with edges that are sections of the lines or n-gon edges.
%H A320431 R. J. Mathar, <a href="/A320431/a320431.pdf">OEIS A320431</a>
%H A320431 Thomas Young, R. J. Mathar, <a href="/A320431/a320431_1.pdf">The surfer and the hut: a polygon dissection</a> (2019)
%H A320431 <a href="/index/Pol#Poonen">Index to sequences on drawing diagonals in regular polygons</a>
%H A320431 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F A320431 a(2n) = 2*n^2+2*n+1 = A001844(n), n>1. a(2n+1) = 8*n^2-1 = A157914(n), n>1. - Thomas Young (tyoung(AT)district16.org), Nov 11 2017
%F A320431 G.f.: x^3 +x^4 -x^5*(31+13*x-22*x^2-14*x^3+7*x^4+5*x^5) / ( (x-1)^3*(1+x)^3 ). - _R. J. Mathar_, Jan 21 2019
%F A320431 a(n) = 1+n*A064680(n-2), n>=5. - _R. J. Mathar_, Jan 21 2019
%Y A320431 Cf. A165217, A320422
%K A320431 nonn,easy
%O A320431 3,3
%A A320431 _R. J. Mathar_, Jan 08 2019