A179259 Arises in covering a graph by forests and a matching.

2, 12, 52, 140, 294, 532, 872, 1332, 1930, 2684, 3612, 4732, 6062, 7620, 9424, 11492, 13842, 16492, 19460, 22764, 26422, 30452, 34872, 39700, 44954, 50652, 56812, 63452, 70590, 78244, 86432, 95172, 104482, 114380, 124884, 136012, 147782, 160212, 173320, 187124

Offset

0,1

Comments

Kaiser proves that for any positive integer k and for epsilson = 1/((k+2)*(3*k^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

Tomás Kaiser, Mickaël Montassier, and André Raspaud, Covering a graph by forests and a matching, arXiv:1007.0316 [math.CO], 2010.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = (n+2)*(3*n^2+1).

From R. J. Mathar, Jul 08 2010: (Start)

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). (End)

E.g.f.: (2 + 10*x + 15*x^2 + 3*x^3)*exp(x). - Elmo R. Oliveira, Jul 06 2026

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 *)
(* Alternative: *)
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

Sequence in context: A043007 A300572 A176580 * A261474 A350653 A080675

Adjacent sequences: A179256 A179257 A179258 * A179260 A179261 A179262

Keyword

easy,nonn,changed

Author

Jonathan Vos Post, Jul 05 2010

Status

approved