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A097075 Expansion of (1-x-x^2)/(1-x-3*x^2-x^3). 5
1, 0, 2, 3, 9, 20, 50, 119, 289, 696, 1682, 4059, 9801, 23660, 57122, 137903, 332929, 803760, 1940450, 4684659, 11309769, 27304196, 65918162, 159140519, 384199201, 927538920, 2239277042, 5406093003, 13051463049, 31509019100, 76069501250 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Counts closed walks of length n at a vertex of a triangle, to which a loop has been added at one of the other vertices.

a(n) is the top left entry of the n-th power of the 3X3 matrix [0, 1, 1; 1, 1, 1; 1, 1, 0] or of the 3X3 matrix [0, 1, 1; 1, 0, 1; 1, 1, 1].

LINKS

Table of n, a(n) for n=0..30.

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) = (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).

a(n) = (-1)^n/2 + sum(k=0..floor(n/2), binomial(n, 2*k)*2^k)/2.

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

PROG

(PARI) Vec((1-x-x^2)/(1-x-3*x^2-x^3) + O(x^50)) \\ Michel Marcus, Mar 25 2014

CROSSREFS

Cf. A000129, A051927, A097076.

Sequence in context: A121908 A231368 A245123 * A036673 A111189 A001004

Adjacent sequences:  A097072 A097073 A097074 * A097076 A097077 A097078

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jul 22 2004

STATUS

approved

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Last modified October 16 09:13 EDT 2018. Contains 316261 sequences. (Running on oeis4.)