

A096976


Number of walks of length n on P_3 plus a loop at the end.


7



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

Table of n, a(n) for n=0..38.
Robin Chapman and Nicholas C. Singer, Eigenvalues of a bidiagonal matrix, Amer. Math. Monthly, 111 (2004) 441
L. E. Jeffery, Unitprimitive matrices
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



