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 A097076 Expansion of x/(1 - x - 3x^2 - x^3). 11
 0, 1, 1, 4, 8, 21, 49, 120, 288, 697, 1681, 4060, 9800, 23661, 57121, 137904, 332928, 803761, 1940449, 4684660, 11309768, 27304197, 65918161, 159140520, 384199200, 927538921, 2239277041, 5406093004, 13051463048, 31509019101, 76069501249, 183648021600 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Counts walks of length n between two vertices of a triangle, when a loop has been added at the third vertex. a(n) is the center term of the 3 X 3 matrix [0,1,0; 0,0,1; 1,3,1]^n. - Gary W. Adamson, May 30 2008 Starting (1, 1, 4, 8, 21, ...) = row sums of triangle A157898. - Gary W. Adamson, Mar 08 2009 Convolution of Pell(n)=A000129(n) and (-1)^n. - Paul Barry, Oct 22 2009 LINKS J. Bodeen, S. Butler, T. Kim, X. Sun and S. Wang, Tiling a strip with triangles, El. J. Combinat. 21 (1) (2014) P1.7. M. Shattuck, Combinatorial Proofs of Some Formulas for Triangular Tilings, Journal of Integer Sequences, 17 (2014), #14.5.5. Index entries for linear recurrences with constant coefficients, signature (1,3,1) FORMULA a(n) = (1+sqrt(2))^n/4 + (1-sqrt(2))^n/4 - (-1)^n/2. a(n) = a(n-1) + 3*a(n-2) + a(n-3). [corrected by Paul Curtz, Mar 04 2008] a(n) = (Sum_{k=0..floor(n/2)} binomial(n, 2*k)*2^k)/2 - (-1)^n/2. a(n) = A001333(n)/2 - (-1)^n/2. a(n) = Sum_{k=0..n} (-1)^k*Pell(n-k). - Paul Barry, Oct 22 2009 G.f.: -x / ( (1+x)*(x^2+2*x-1) ). - R. J. Mathar, Jul 06 2011 a(n) + a(n+1) = A000129(n+1). - R. J. Mathar, Jul 06 2011 MATHEMATICA 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 *) CROSSREFS Cf. A000129, A001333, A051927, A097075, A157898. Sequence in context: A244583 A261031 A077921 * A003608 A248423 A319565 Adjacent sequences:  A097073 A097074 A097075 * A097077 A097078 A097079 KEYWORD easy,nonn AUTHOR Paul Barry, Jul 22 2004 STATUS approved

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Last modified July 6 12:26 EDT 2022. Contains 355110 sequences. (Running on oeis4.)