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

3,3

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).

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.

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]

Prog

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

Crossrefs

Even-indexed terms of A000096 interleaved with A005563.

Cf. A002378, A211379.

Sequence in context: A083594 A178497 A364888 * A213750 A213774 A167583

Adjacent sequences: A367201 A367202 A367203 * A367205 A367206 A367207

Keyword

nonn,easy

Author

Paolo Xausa, Nov 10 2023

Status

approved