

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
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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 7fold rotational symmetry. Let A_{7,1} be the 3 X 3 unitprimitive 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 nth 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, Unitprimitive 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. : (1xx^2)/(1x2x^2+x^3); a(n)=a(n1)+2a(n2)a(n3).
a(n) = 5a(n2)6a(n4)+a(n6).  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



