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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. - _Gary W. Adamson_, Mar 08 2009

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

%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 M. Shattuck, <a href="https://www.emis.de/journals/JIS/VOL17/Shattuck/shattuck8.html">Combinatorial Proofs of Some Formulas for Triangular Tilings</a>, Journal of Integer Sequences, 17 (2014), #14.5.5.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,1)

%F a(n) = (1+sqrt(2))^n/4 + (1-sqrt(2))^n/4 - (-1)^n/2.

%F a(n) = a(n-1) + 3*a(n-2) + a(n-3). [corrected by Paul Curtz, Mar 04 2008]

%F a(n) = (Sum_{k=0..floor(n/2)} binomial(n, 2*k)*2^k)/2 - (-1)^n/2.

%F a(n) = A001333(n)/2 - (-1)^n/2.

%F a(n) = Sum_{k=0..n} (-1)^k*Pell(n-k). - _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 November 14 08:38 EST 2018. Contains 317175 sequences. (Running on oeis4.)