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A052921 Expansion of (1-x)/(1-3*x+2*x^2-x^3). 8
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; text; internal format)
OFFSET

0,2

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

Partial sums of A034943 starting (1, 1, 2, 5, 12, 28, 65,...). - Gary W. Adamson, Feb 15 2012

a(n) is the number of n (decimal) digit integers x such that all digits of x are odd and all digits of 6x are even. - Robert Israel, Apr 17 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 905

Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

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)}

a(n) = 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

a(n) = A000931(3*n + 5). - Michael Somos, Sep 18 2012

EXAMPLE

1 + 2*x + 4*x^2 + 9*x^3 + 21*x^4 + 49*x^5 + 114*x^6 + 265*x^7 + ...

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

MATHEMATICA

a=-1; b=0; c=1; lst={}; Do[AppendTo[lst, a+=b]; b+=c; c+=a, {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 20 2009 *)

LinearRecurrence[{3, -2, 1}, {1, 2, 4}, 40] (* Vincenzo Librandi, Feb 14 2012 *)

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

CROSSREFS

Cf. A034943.

Cf. A097550, A137531.

Sequence in context: A101891 A119967 A266232 * A219150 A275864 A018905

Adjacent sequences:  A052918 A052919 A052920 * A052922 A052923 A052924

KEYWORD

nonn,easy

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

STATUS

approved

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Last modified August 16 00:25 EDT 2018. Contains 313782 sequences. (Running on oeis4.)