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A052921
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Expansion of (1-x)/(1-3x+2x^2-x^3).
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7
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1, 2, 4, 9, 21, 49, 114, 265, 616, 1432, 3329, 7739, 17991, 41824, 97229, 226030, 525456, 1221537, 2839729, 6601569, 15346786, 35676949, 82938844, 192809420, 448227521, 1042002567, 2422362079, 5631308624, 13091204281, 30433357674
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The Ca3 sums, see A180662, of triangle A065941 equal the terms of this sequence. [Johannes W. Meijer, Aug 16 2011]
First differences of A095263. - R. J. Mathar, Nov 23 2011
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 905
Index to sequences with linear recurrences with constant coefficients, signature (3,-2,1).
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FORMULA
| G.f.: (-1+x)/(-1+3*x-2*x^2+x^3)
Recurrence: {a(0)=1, a(2)=4, a(1)=2, a(n)-2*a(n+1)+3*a(n+2)-a(n+3)}
Sum(1/23*(8-5*_alpha+7*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(-1+3*_Z-2*_Z^2+_Z^3))
Binomial transform of the Padovan sequence A000931(n+5). a(n) = sum{k=0..n+1, C(n+k+1, n-2*k)}. - Paul Barry, Jun 21 2004
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MAPLE
| spec := [S, {S=Sequence(Union(Z, Z, Prod(Sequence(Z), Z, Z, Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..29);
A052921 := proc(n): add(binomial(n+k+1, n-2*k), k=0..n+1) end: seq(A052921(n), n=0..29); # [Johannes W. Meijer, Aug 16 2011]
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MATHEMATICA
| a=-1; b=0; c=1; lst={}; Do[AppendTo[lst, a+=b]; b+=c; c+=a, {n, 5!}]; lst [From Vladimir Orlovsky, Jan 20 2009]
LinearRecurrence[{3, -2, 1}, {1, 2, 4}, 40] (* Vincenzo Librandi, Feb 14 2012 *)
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PROG
| (MAGMA) I:=[1, 2, 4]; [n le 3 select I[n] else 3*Self(n-1)-2*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 14 2012
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CROSSREFS
| Cf. A034943.
Cf. A097550, A137531.
Sequence in context: A182904 A101891 A119967 * A018905 A024537 A171842
Adjacent sequences: A052918 A052919 A052920 * A052922 A052923 A052924
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KEYWORD
| nonn,easy,changed
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AUTHOR
| encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 05 2000
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