A062346 Consider 2n tennis players; a(n) is the number of matches needed to let every possible pair play each other.

3, 45, 210, 630, 1485, 3003, 5460, 9180, 14535, 21945, 31878, 44850, 61425, 82215, 107880, 139128, 176715, 221445, 274170, 335790, 407253, 489555, 583740, 690900, 812175, 948753, 1101870, 1272810, 1462905, 1673535, 1906128, 2162160, 2443155

Offset

2,1

Comments

Number of matchings of size two (edges) in a complete graph on 2n vertices.

Links

Vincenzo Librandi, Table of n, a(n) for n = 2..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = n*(4*n^3 - 12*n^2 + 11*n - 3)/2. - Swapnil P. Bhatia (sbhatia(AT)cs.unh.edu), Jul 20 2006

a(n+1) = (2*n+2)*(2*n+1)*(2*n)*(2*n-1)/8. - James Mahoney, Oct 19 2011

G.f.: 3*x^2*(1 + 10*x + 5*x^2)/(1 - x)^5. - Vincenzo Librandi, Oct 13 2013

a(n) = binomial(2*n^2-3*n+1, 2). - Wesley Ivan Hurt, Oct 14 2013

a(n) = A014105(n-1)*(A014105(n-1)-1)/2. - Bruno Berselli, Dec 28 2016

Example

a(2)=3: given players a,b,c,d, the matches needed are (ab,cd), (ac,bd), (ad,bc).
For example, for the K_4 on vertices {0,1,2,3} the possible matchings of size two are: {{0,1}, {2,3}}, {{0,3},{1,2}} and {{0,2},{1,3}}.

Maple

A062346:=n->n*(n-1)*(2*n-3)*(2*n-1)/2; seq(A062346(k), k=2..100); # Wesley Ivan Hurt, Oct 14 2013

Mathematica

CoefficientList[Series[3 (1 + 10 x + 5 x^2)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 13 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {3, 45, 210, 630, 1485}, 40] (* Harvey P. Dale, Nov 22 2022 *)

Prog

(PARI) a(n) = n*(n-1)*(2*n-3)*(2*n-1)/2; \\ Joerg Arndt, Oct 13 2013
(Magma) [n*(n-1)*(2*n-3)*(2*n-1)/2: n in [2..40]]; // Vincenzo Librandi, Oct 13 2013

Crossrefs

Cf. A014105.

Sequence in context: A062270 A069955 A289193 * A359860 A002682 A073595

Adjacent sequences: A062343 A062344 A062345 * A062347 A062348 A062349

Keyword

nonn,easy

Author

Michel ten Voorde, Jul 06 2001

Extensions

More terms from Swapnil P. Bhatia (sbhatia(AT)cs.unh.edu), Jul 20 2006

Status

approved