%I #10 Jul 02 2024 02:54:54
%S 0,0,0,0,1,1,2,3,4,6,8,9,12,14,16,20,23,25,30,33,36,42,46,49,56,60,64,
%T 72,77,81,90,95,100,110,116,121,132,138,144,156,163,169,182,189,196,
%U 210,218,225,240,248,256,272,281,289,306,315,324,342,352,361
%N a(n) = ceiling(1/2 (n - 3 - ceiling((n - 3)/3))*ceiling((n - 3)/3)).
%C a(n) is the rectilinear local crossing number of the complete graph K_n except for n = 8, 14.
%H B. M. Ábrego and S. Fernández-Merchant, <a href="https://doi.org/10.1016/j.jcta.2017.04.003">The Rectilinear Local Crossing Number of K_n</a>, J. Combin. Th. Ser. A, 151 (2017), 131-145.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RectilinearLocalCrossingNumber.html">Rectilinear Local Crossing Number</a>.
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1,0,1,-1,0,-1,1).
%F G.f.: x^5*(-1-x^2-x^4-x^5)/((-1+x)^3*(1+x+x^2)^2*(1+x^3)).
%F a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-6) - a(n-7) - a(n-9} + a(n-10).
%t Table[Ceiling[1/2 (n - 3 - Ceiling[(n - 3)/3]) Ceiling[(n - 3)/3]], {n, 20}]
%t Table[(89 - 9 (-1)^n + 6 n (-11 + 2 n) + 9 Cos[n Pi/3] + (19 - 6 n) Cos[2 n Pi/3] - 9 Sqrt[3] Sin[n Pi/3] + Sqrt[3] (-21 + 2 n) Sin[2 n Pi/3])/108, {n, 20}]
%t LinearRecurrence[{1, 0, 1, -1, 0, 1, -1, 0, -1, 1}, {0, 0, 0, 0, 1, 1, 2, 3, 4, 6}, 20]
%t CoefficientList[Series[x^4 (-1 - x^2 - x^4 - x^5)/((-1 + x)^3 (1 + x + x^2)^2 (1 + x^3)), {x, 0, 20}], x]
%K nonn,easy
%O 1,7
%A _Eric W. Weisstein_, Jul 01 2024