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.

1, 1, 31, 13, 71, 25, 127, 41, 199, 61, 287, 85, 391, 113, 511, 145, 647, 181, 799, 221, 967, 265, 1151, 313, 1351, 365, 1567, 421, 1799, 481, 2047, 545, 2311, 613, 2591, 685, 2887, 761, 3199, 841, 3527, 925, 3871, 1013, 4231, 1105, 4607, 1201, 4999, 1301, 5407, 1405, 5831, 1513, 6271, 1625, 6727, 1741

Offset: 3

Author

R. J. Mathar, Jan 08 2019

Keywords

• nonn
• easy

Comments

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.

Links

• R. J. Mathar, OEIS A320431
• Thomas Young, R. J. Mathar, The surfer and the hut: a polygon dissection (2019)
• Index to sequences on drawing diagonals in regular polygons
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A165217, A320422

Formula

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

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

a(n) = 1+n*A064680(n-2), n>=5. - R. J. Mathar, Jan 21 2019