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%I #38 Sep 27 2022 09:01:58
%S 0,0,0,2,4,6,10,14,18,24,30,36,44,52,60,70,80,90,102,114,126,140,154,
%T 168,184,200,216,234,252,270,290,310,330,352,374,396,420,444,468,494,
%U 520,546,574,602,630,660,690,720,752,784,816,850,884,918,954,990,1026
%N Projective plane crossing number of K_{4,n}.
%C From _Gus Wiseman_, Oct 15 2020: (Start)
%C Also the number of 3-part compositions of n that are neither strictly increasing nor weakly decreasing. The set of numbers k such that row k of A066099 is such a composition is the complement of A333255 (strictly increasing) and A114994 (weakly decreasing) in A014311 (triples). The a(4) = 2 through a(9) = 14 compositions are:
%C (1,1,2) (1,1,3) (1,1,4) (1,1,5) (1,1,6) (1,1,7)
%C (1,2,1) (1,2,2) (1,3,2) (1,3,3) (1,4,3) (1,4,4)
%C (1,3,1) (1,4,1) (1,4,2) (1,5,2) (1,5,3)
%C (2,1,2) (2,1,3) (1,5,1) (1,6,1) (1,6,2)
%C (2,3,1) (2,1,4) (2,1,5) (1,7,1)
%C (3,1,2) (2,2,3) (2,2,4) (2,1,6)
%C (2,3,2) (2,3,3) (2,2,5)
%C (2,4,1) (2,4,2) (2,4,3)
%C (3,1,3) (2,5,1) (2,5,2)
%C (4,1,2) (3,1,4) (2,6,1)
%C (3,2,3) (3,1,5)
%C (3,4,1) (3,2,4)
%C (4,1,3) (3,4,2)
%C (5,1,2) (3,5,1)
%C (4,1,4)
%C (4,2,3)
%C (5,1,3)
%C (6,1,2)
%C (End)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ProjectivePlaneCrossingNumber.html">Projective Plane Crossing Number</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F a(n) = floor(n/3)*(2n-3(floor(n/3)+1)).
%F a(n) = ceiling(n^2/3) - n. - _Charles R Greathouse IV_, Jun 06 2013
%F G.f.: -2*x^4 / ((x-1)^3*(x^2+x+1)). - _Colin Barker_, Jun 06 2013
%F a(n) = floor((n - 1)(n - 2) / 3). - _Christopher Hunt Gribble_, Oct 13 2009
%F a(n) = 2*A001840(n-3). - _R. J. Mathar_, Jul 21 2015
%F a(n) = A000217(n-2) - A001399(n-6) - A001399(n-3). - _Gus Wiseman_, Oct 15 2020
%F Sum_{n>=4} 1/a(n) = 10/3 - Pi/sqrt(3). - _Amiram Eldar_, Sep 27 2022
%t Table[Floor[((n - 2)^2 + (n - 2))/3], {n, 1, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 31 2012 *)
%t Table[Ceiling[n^2/3] - n, {n, 20}] (* _Eric W. Weisstein_, Sep 07 2018 *)
%t Table[(3 n^2 - 9 n + 4 - 4 Cos[2 n Pi/3])/9, {n, 20}] (* _Eric W. Weisstein_, Sep 07 2018 *)
%t LinearRecurrence[{2, -1, 1, -2, 1}, {0, 0, 0, 2, 4, 6}, 20] (* _Eric W. Weisstein_, Sep 07 2018 *)
%t CoefficientList[Series[-2 x^3/((-1 + x)^3 (1 + x + x^2)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 07 2018 *)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],!Less@@#&&!GreaterEqual@@#&]],{n,15}] (* _Gus Wiseman_, Oct 15 2020 *)
%o (PARI) a(n)=(n-1)*(n-2)\3 \\ _Charles R Greathouse IV_, Jun 06 2013
%Y A007997 counts the complement.
%Y A337482 counts these compositions of any length.
%Y A337484 is the non-strict/non-strict version.
%Y A000009 counts strictly increasing compositions, ranked by A333255.
%Y A000041 counts weakly decreasing compositions, ranked by A114994.
%Y A001523 counts unimodal compositions (strict: A072706).
%Y A007318 and A097805 count compositions by length.
%Y A032020 counts strict compositions, ranked by A233564.
%Y A225620 ranks weakly increasing compositions.
%Y A333149 counts neither increasing nor decreasing strict compositions.
%Y A333256 ranks strictly decreasing compositions.
%Y A337483 counts 3-part weakly increasing or weakly decreasing compositions.
%Y Cf. A000217, A001399, A001840, A059204, A072705, A115981, A329398, A332578, A332831, A332833, A332834, A332874, A333147, A333190.
%K nonn,easy
%O 1,4
%A _Eric W. Weisstein_, Mar 02 2007
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