OFFSET
0,1
COMMENTS
Kaiser proves that for any positive integer k and for epsilson = 1/((k+2)(3k^2+1)), the edges of any graph whose fractional arboricity is at most k + epsilon can be decomposed into $k$ forests and a matching.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tomas Kaiser, Mickael Montassier, Andre Raspaud, Covering a graph by forests and a matching, Jul 02, 2010.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = (n+2)*(3*n^2+1).
G.f.: -2*(-1-2*x-8*x^2+2*x^3)/(x-1)^4. a(n) = +4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). a(n) = 0 (mod 2). [From R. J. Mathar, Jul 08 2010]
EXAMPLE
a(4) = (4+2)*(3*4^2+1) = 294.
MATHEMATICA
CoefficientList[Series[-2*(-1-2*x-8*x^2+2*x^3)/(x-1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 04 2012 *)
LinearRecurrence[{4, -6, 4, -1}, {2, 12, 52, 140}, 40] (* Harvey P. Dale, Apr 01 2019 *)
PROG
(Magma) I:=[2, 12, 52, 140]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jul 04 2012
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Jul 05 2010
STATUS
approved