A117717 Maximal number of regions obtained by a straight line drawing of the complete bipartite graph K_{n,n}.

0, 2, 13, 45, 116, 250, 477, 833, 1360, 2106, 3125, 4477, 6228, 8450, 11221, 14625, 18752, 23698, 29565, 36461, 44500, 53802, 64493, 76705, 90576, 106250, 123877, 143613, 165620, 190066, 217125, 246977, 279808, 315810, 355181, 398125, 444852, 495578, 550525

Offset: 1

Author

Patricia A. Carey and Anant Godbole, Apr 13 2006

Keywords

• nonn
• easy

Comments

This sequence is in the same spirit as A000127 where a formula is given for the maximal number of regions obtained by a straight line drawing of the complete graph K_n with the vertices located on the perimeter of a circle. This yields the often quoted sequence A000127.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000127.

Programs

Magma

[(n-1)^2*(n^2+4)/4: n in [1..40]]; // Vincenzo Librandi, Sep 09 2011

Maple

A117717 := proc(n)
    (n-1)^2*(n^2+4)/4 ;
end proc:
seq(A117717(n),n=1..10) ; # R. J. Mathar, Sep 15 2013

Mathematica

Table[n^2-2n+Binomial[n,2]^2+1,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,2,13,45,116},40] (* Harvey P. Dale, Oct 16 2012 *)

Formula

a(n) = n^2 - 2n + C(n,2)^2 + 1

a(n) = (n-1)^2*(n^2+4)/4. - Vincenzo Librandi, Sep 09 2011

G.f.: x^2*(2+3*x+x^3)/(1-x)^5. - Colin Barker, Feb 15 2012

a(n)=5*a(n-1)-10*a(n-2)+ 10*a(n-3)- 5*a(n-4)+a(n-5), n>5. - Harvey P. Dale, Oct 16 2012

Extensions

More terms from Harvey P. Dale, Oct 16 2012