A080838 - OEIS

%I #56 Sep 19 2024 09:17:18

%S 0,0,0,2,5,12,21,36,54,80,110,150,195,252,315,392,476,576,684,810,945,

%T 1100,1265,1452,1650,1872,2106,2366,2639,2940,3255,3600,3960,4352,

%U 4760,5202,5661,6156,6669,7220,7790,8400,9030,9702,10395,11132,11891

%N Orchard crossing number of complete bipartite graph K_{1,n}.

%C Also the minimum number of transitive triples in a tournament on n nodes, i.e., a(n) = C(n,3) - A006918(n-2). - _Leen Droogendijk_, Nov 10 2014

%C a(n) = the number of binary strings of length n+1 with exactly one pair of adjacent 0's and exactly two pairs of adjacent 1's. - _Jeremy Dover_, Jul 07 2016

%H Vincenzo Librandi, <a href="/A080838/b080838.txt">Table of n, a(n) for n = 1..1000</a>

%H D. Garber, <a href="http://arXiv.org/abs/math.CO/0303317">The Orchard crossing number of an abstract graph</a>, arXiv:math/0303317 [math.CO], 2003.

%H M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2, 1, -4, 1, 2, -1).

%F a(n) = (n/16) * (2*n^2 - 8*n + 7 + (-1)^n).

%F G.f.: (x^5 + 2*x^4) / (1-x)^4 / (1+x)^2.

%F For n odd, a(n) = A060423(n). - _Gerald McGarvey_, Sep 14 2008

%t CoefficientList[Series[(x^4 + 2 x^3) / (1 - x)^4 / (1 + x)^2, {x, 0, 40}], x] (* _Vincenzo Librandi_, May 17 2013 *)

%t Table[n/16*(2 n^2 - 8 n + 7 + (-1)^n), {n, 47}] (* _Michael De Vlieger_, Aug 01 2016 *)

%o (PARI) for(n=1,100,print1(if(n%2,n*(n-1)*(n-3)/8,n*(n-2)^2/8)","))

%o (Magma) [n/16*(2*n^2 - 8*n + 7 + (-1)^n): n in [1..50]]; // _Vincenzo Librandi_, May 17 2013

%Y Third column of A274228. - _Jeremy Dover_, Jul 07 2016

%Y Essentially partial sums of A211539.

%K nonn,easy

%O 1,4

%A _Ralf Stephan_, Mar 28 2003