cp's OEIS Frontend

A211380 Number of pairs of intersecting diagonals in the interior and exterior of a regular n-gon.

%I A211380 #23 Feb 16 2025 08:33:17
%S A211380 0,1,5,15,42,94,189,340,572,903,1365,1981,2790,3820,5117,6714,8664,
%T A211380 11005,13797,17083,20930,25386,30525,36400,43092,50659,59189,68745,
%U A211380 79422,91288,104445,118966,134960,152505,171717,192679,215514,240310,267197,296268,327660
%N A211380 Number of pairs of intersecting diagonals in the interior and exterior of a regular n-gon.
%H A211380 Eric Weisstein, <a href="https://mathworld.wolfram.com/RegularPolygonDivisionbyDiagonals.html">Regular Polygon Division by Diagonals</a> (MathWorld).
%H A211380 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).
%F A211380 a(n) = 1/8*n*(n^3-11*n^2+43*n-58) for n even;
%F A211380 a(n) = 1/8*n*(n-3)*(n^2-8*n+19) for n odd.
%F A211380 a(n) = A176145(n) - A211379(n).
%F A211380 G.f.: x^4*(2*x^5-3*x^4-7*x^3-x^2-2*x-1) / ((x-1)^5*(x+1)^2). [_Colin Barker_, Feb 14 2013]
%p A211380 a:=n->piecewise(n mod 2 = 0,1/8*n*(n^3-11*n^2+43*n-58),n mod 2 = 1,1/8*n*(n-3)*(n^2-8*n+19),0);
%t A211380 Drop[CoefficientList[Series[x^4(2x^5-3x^4-7x^3-x^2-2x-1)/((x-1)^5(x+1)^2),{x,0,50}],x],3] (* or *) LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,1,5,15,42,94,189},50] (* _Harvey P. Dale_, Dec 03 2022 *)
%o A211380 (Python)
%o A211380 def A211380(n): return n*(n*(n*(n-11)+43)-58+(n&1))>>3 # _Chai Wah Wu_, Nov 22 2023
%Y A211380 Cf. A176145, A211379.
%K A211380 nonn,easy
%O A211380 3,3
%A A211380 _Martin Renner_, Feb 07 2013