A367204 Number of diagonals in a regular n-gon that are parallel to an edge.

0, 0, 5, 3, 14, 8, 27, 15, 44, 24, 65, 35, 90, 48, 119, 63, 152, 80, 189, 99, 230, 120, 275, 143, 324, 168, 377, 195, 434, 224, 495, 255, 560, 288, 629, 323, 702, 360, 779, 399, 860, 440, 945, 483, 1034, 528, 1127, 575, 1224, 624, 1325, 675, 1430, 728, 1539, 783

Offset: 3

Author

Paolo Xausa, Nov 10 2023

Keywords

• nonn
• easy

Comments

A diagonal is parallel to an edge if and only if, on at least one side of the diagonal, there is an odd number of edges.

If n is odd, all of the diagonals of the n-gon are parallel to an edge.

Links

• Paolo Xausa, Table of n, a(n) for n = 3..10000
• Thomas Andrews, Number of Parallel/Not Parallel Diagonals of a Regular Polygon, Mathematics Stack Exchange, 2014.
• Paolo Xausa, Illustration of first terms.
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Even-indexed terms of A000096 interleaved with A005563.

Cf. A002378, A211379.

Programs

Mathematica

LinearRecurrence[{0,3,0,-3,0,1},{0,0,5,3,14,8},100] (* or *)
A367204[n_]:=If[OddQ[n],n(n-3)/2,n(n-4)/4];Array[A367204,100,3]

Python

def A367204(n): return n*(n-3)>>1 if n&1 else n*(n-4)>>2 # Chai Wah Wu, Nov 22 2023

Formula

a(n) = n(n-3)/2 = A000096(n-3) if n is odd;

a(n) = n(n-4)/4 = A005563((n-4)/2) = A000096(n-3) - A002378(n/2-1) if n is even.

Sum_{n>=5} 1/a(n) = 59/36 - 2*log(2)/3. - Amiram Eldar, Dec 06 2024