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A096976 Number of walks of length n on P_3 plus a loop at the end. 8
1, 0, 1, 0, 2, 1, 5, 5, 14, 19, 42, 66, 131, 221, 417, 728, 1341, 2380, 4334, 7753, 14041, 25213, 45542, 81927, 147798, 266110, 479779, 864201, 1557649, 2806272, 5057369, 9112264, 16420730, 29587889, 53317085, 96072133, 173118414, 311945595, 562110290 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Counts closed walks of length n at the start of P_3 to which a loop has been added at the other extremity. a(n+1) counts walks between the first node and the last. Let A be the adjacency matrix of the graph P_3 with a loop added at the end. A is a 'reverse Jordan matrix' [0,0,1;0,1,1;1,1,0]. a(n) is obtained by taking the (1,1) element of A^n.

Sequence is also related to matrices associated with rhombus substitution tilings showing 7-fold rotational symmetry. Let A_{7,1} be the 3 X 3 unit-primitive matrix (see [Jeffery]) A_{7,1}=[0,1,0; 1,0,1; 0,1,1]; then a(n)=[A_{7,1}^n]_(1,1). - L. Edson Jeffery, Jan 05 2012

a(n+2) is the (1,1) element of the n-th power of each of the two 3 X 3 matrices: [0,1,1; 1,0,0; 1,0,1], [0,1,1; 1,1,0; 1,0,0]. - Christopher Hunt Gribble, Apr 03 2014

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..3914

Robin Chapman and Nicholas C. Singer, Eigenvalues of a bidiagonal matrix, Amer. Math. Monthly, 111 (2004) 441.

L. E. Jeffery, Unit-primitive matrices

Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).

Index entries for linear recurrences with constant coefficients, signature (1,2,-1).

FORMULA

G.f. : (1-x-x^2)/(1-x-2x^2+x^3); a(n)=a(n-1)+2a(n-2)-a(n-3).

a(n) = 5a(n-2)-6a(n-4)+a(n-6). - Floor van Lamoen, Nov 02 2005

MATHEMATICA

LinearRecurrence[{1, 2, -1}, {1, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

CROSSREFS

Cf. A006053, A052547.

Sequence in context: A167158 A074392 A284428 * A052547 A119245 A128731

Adjacent sequences:  A096973 A096974 A096975 * A096977 A096978 A096979

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jul 16 2004

STATUS

approved

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Last modified November 28 14:27 EST 2020. Contains 338724 sequences. (Running on oeis4.)