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A097076 Expansion of x/(1-x-3x^2-x^3). 10
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) = center term of the 3x3 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 [From Gary W. Adamson, Mar 08 2009]

Convolution of Pell(n)=A000129(n) and (-1)^n. [From Paul Barry, Oct 22 2009]

REFERENCES

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

LINKS

Table of n, a(n) for n=0..31.

J. Bodeen, S. Butler, T. Kim, X. Sun, S. Wang, Tiling a strip with triangles, El. J. Combinat. 21 (1) (2014) P1.7

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

FORMULA

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.

a(n)=sum{k=0..n, (-1)^k*Pell(n-k)}. [From 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, A051927, A097075, A157898.

Sequence in context: A244583 A261031 A077921 * A003608 A248423 A129794

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 February 18 23:16 EST 2018. Contains 299330 sequences. (Running on oeis4.)