%I #37 Nov 22 2023 01:38:36

%S 0,0,5,3,14,8,27,15,44,24,65,35,90,48,119,63,152,80,189,99,230,120,

%T 275,143,324,168,377,195,434,224,495,255,560,288,629,323,702,360,779,

%U 399,860,440,945,483,1034,528,1127,575,1224,624,1325,675,1430,728,1539,783

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

%C 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.

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

%H Paolo Xausa, <a href="/A367204/b367204.txt">Table of n, a(n) for n = 3..10000</a>

%H Thomas Andrews, <a href="https://math.stackexchange.com/q/793833">Number of Parallel/Not Parallel Diagonals of a Regular Polygon</a>, Mathematics Stack Exchange, 2014.

%H Paolo Xausa, <a href="/A367204/a367204_1.pdf">Illustration of first terms</a>.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).

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

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

%t LinearRecurrence[{0,3,0,-3,0,1},{0,0,5,3,14,8},100] (* or *)

%t A367204[n_]:=If[OddQ[n],n(n-3)/2,n(n-4)/4];Array[A367204,100,3]

%o (Python)

%o def A367204(n): return n*(n-3)>>1 if n&1 else n*(n-4)>>2 # _Chai Wah Wu_, Nov 22 2023

%Y Even-indexed terms of A000096 interleaved with A005563.

%Y Cf. A002378, A211379.

%K nonn,easy

%O 3,3

%A _Paolo Xausa_, Nov 10 2023