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A179259
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Arises in covering a graph by forests and a matching.
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1
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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]
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EXAMPLE
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a(4) = (4+2)*(3*4^2+1) = 294.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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