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A097076 Expansion of x/(1-x-3x^2-x^3). 10

%I

%S 0,1,1,4,8,21,49,120,288,697,1681,4060,9800,23661,57121,137904,332928,

%T 803761,1940449,4684660,11309768,27304197,65918161,159140520,

%U 384199200,927538921,2239277041,5406093004,13051463048,31509019101,76069501249,183648021600

%N Expansion of x/(1-x-3x^2-x^3).

%C Counts walks of length n between two vertices of a triangle, when a loop has been added at the third vertex.

%C a(n) = center term of the 3x3 matrix [0,1,0; 0,0,1; 1,3,1]^n - _Gary W. Adamson_, May 30 2008

%C Starting (1, 1, 4, 8, 21,...) = row sums of triangle A157898 [From _Gary W. Adamson_, Mar 08 2009]

%C Convolution of Pell(n)=A000129(n) and (-1)^n. [From _Paul Barry_, Oct 22 2009]

%D M. Shattuck, Combinatorial Proofs of Some Formulas for Triangular Tilings, Journal of Integer Sequences, 17 (2014), #14.5.5.

%H J. Bodeen, S. Butler, T. Kim, X. Sun, S. Wang, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i1p7">Tiling a strip with triangles</a>, El. J. Combinat. 21 (1) (2014) P1.7

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,1)

%F a(n)=a(n)=(1+sqrt(2))^n/4+(1-sqrt(2))^n/4-(-1)^n/2; a(n)=a(n-1)+3a(n-2)+a(n-3) [corrected by Paul Curtz, Mar 04 2008]; a(n)=sum{k=0..floor(n/2), binomial(n, 2k)2^k}/2-(-1)^n/2. a(n)=A001333(n)/2-(-1)^n/2.

%F a(n)=sum{k=0..n, (-1)^k*Pell(n-k)}. [From _Paul Barry_, Oct 22 2009]

%F G.f. -x / ( (1+x)*(x^2+2*x-1) ). - R. J. Mathar, Jul 06 2011

%F a(n)+a(n+1) = A000129(n+1). - R. J. Mathar, Jul 06 2011

%t CoefficientList[Series[x/(1-x-3x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{1,3,1},{0,1,1},40] (* _Vladimir Joseph Stephan Orlovsky_, Jan 30 2012 *)

%Y Cf. A000129, A051927, A097075, A157898.

%K easy,nonn

%O 0,4

%A _Paul Barry_, Jul 22 2004

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Last modified February 19 22:04 EST 2018. Contains 299357 sequences. (Running on oeis4.)