A080838 Orchard crossing number of complete bipartite graph K_{1,n}.

0, 0, 0, 2, 5, 12, 21, 36, 54, 80, 110, 150, 195, 252, 315, 392, 476, 576, 684, 810, 945, 1100, 1265, 1452, 1650, 1872, 2106, 2366, 2639, 2940, 3255, 3600, 3960, 4352, 4760, 5202, 5661, 6156, 6669, 7220, 7790, 8400, 9030, 9702, 10395, 11132, 11891

Offset: 1

Author

Ralf Stephan, Mar 28 2003

Keywords

• nonn
• easy

Comments

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

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• D. Garber, The Orchard crossing number of an abstract graph, arXiv:math/0303317 [math.CO], 2003.
• M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
• Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Crossrefs

Third column of A274228. - Jeremy Dover, Jul 07 2016

Essentially partial sums of A211539.

Programs

Magma

[n/16*(2*n^2 - 8*n + 7 + (-1)^n): n in [1..50]]; // Vincenzo Librandi, May 17 2013

Mathematica

CoefficientList[Series[(x^4 + 2 x^3) / (1 - x)^4 / (1 + x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, May 17 2013 *)
Table[n/16*(2 n^2 - 8 n + 7 + (-1)^n), {n, 47}] (* Michael De Vlieger, Aug 01 2016 *)

PARI

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

Formula

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

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

For n odd, a(n) = A060423(n). - Gerald McGarvey, Sep 14 2008