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%I #18 Sep 08 2022 08:45:54
%S 2,12,52,140,294,532,872,1332,1930,2684,3612,4732,6062,7620,9424,
%T 11492,13842,16492,19460,22764,26422,30452,34872,39700,44954,50652,
%U 56812,63452,70590,78244,86432,95172,104482,114380,124884,136012,147782
%N Arises in covering a graph by forests and a matching.
%C 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.
%H Vincenzo Librandi, <a href="/A179259/b179259.txt">Table of n, a(n) for n = 0..1000</a>
%H Tomas Kaiser, Mickael Montassier, Andre Raspaud, <a href="http://arxiv.org/abs/1007.0316">Covering a graph by forests and a matching</a>, Jul 02, 2010.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = (n+2)*(3*n^2+1).
%F 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]
%e a(4) = (4+2)*(3*4^2+1) = 294.
%t CoefficientList[Series[-2*(-1-2*x-8*x^2+2*x^3)/(x-1)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jul 04 2012 *)
%t LinearRecurrence[{4,-6,4,-1},{2,12,52,140},40] (* _Harvey P. Dale_, Apr 01 2019 *)
%o (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
%K easy,nonn
%O 0,1
%A _Jonathan Vos Post_, Jul 05 2010