A211379 Number of pairs of parallel diagonals in a regular n-gon.

0, 0, 0, 3, 7, 16, 27, 45, 66, 96, 130, 175, 225, 288, 357, 441, 532, 640, 756, 891, 1035, 1200, 1375, 1573, 1782, 2016, 2262, 2535, 2821, 3136, 3465, 3825, 4200, 4608, 5032, 5491, 5967, 6480, 7011, 7581, 8170, 8800, 9450, 10143, 10857, 11616, 12397, 13225

Offset

3,4

Links

Paolo Xausa, Table of n, a(n) for n = 3..10000

Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Formula

a(n) = (1/2)*n*(binomial(n/2-1,2) + binomial(n/2-2,2)) = (1/8)*n*(n-4)^2 for n even;

a(n) = n*binomial((n-3)/2,2) = (1/8)*n*(n-3)*(n-5) for n odd.

G.f.: -x^6*(x^2-x-3) / ((x-1)^4*(x+1)^2). - Colin Barker, Feb 14 2013

Example

a(6) = 3 since by numbering the vertices from 1 to 6 there are three pairs of parallel diagonals, i.e., {[1, 3], [4, 6]}, {[1, 5], [2, 4]}, {[2, 6], [3, 5]}.
a(7) = 7 since there are the seven pairs {[1, 3], [4, 7]}, {[1, 4], [5, 7]}, {[1, 5], [2, 4]}, {[1, 6], [2, 5]}, {[2, 6], [3, 5]}, {[2, 7], [3, 6]}, {[3, 7], [4, 6]}.
a(8) = 16 since there are the sixteen pairs {[1, 3], [4, 8]}, {[1, 3], [5, 7]}, {[1, 4], [5, 8]}, {[1, 5], [2, 4]}, {[1, 5], [6, 8]}, {[1, 6], [2, 5]}, {[1, 7], [2, 6]}, {[1, 7], [3, 5]}, {[2, 4], [6, 8]}, {[2, 6], [3, 5]}, {[2, 7], [3, 6]}, {[2, 8], [3, 7]}, {[2, 8], [4, 6]}, {[3, 7], [4, 6]}, {[3, 8], [4, 7]}, {[4, 8], [5, 7]}.

Maple

a:=n->piecewise(n mod 2 = 0, 1/8*n*(n-4)^2, n mod 2 = 1, 1/8*n*(n-3)*(n-5), 0);

Mathematica

A211379[n_]:=n/8If[OddQ[n], (n-3)(n-5), (n-4)^2]; Array[A211379, 100, 3] (* or *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 0, 0, 3, 7, 16}, 100] (* Paolo Xausa, Nov 21 2023 *)

Prog

(Python)
def A211379(n): return n*(n*(n-8)+16-(n&1))>>3 # Chai Wah Wu, Nov 22 2023

Crossrefs

Cf. A000096, A367204.

Sequence in context: A036666 A218359 A117491 * A373104 A213180 A110585

Adjacent sequences: A211376 A211377 A211378 * A211380 A211381 A211382

Keyword

nonn,easy

Author

Martin Renner, Feb 07 2013

Status

approved