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