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A096976
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Number of walks of length n on P_3 plus a loop at the end.
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6
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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; internal format)
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OFFSET
| 0,5
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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
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REFERENCES
| R. Chapman, Eigenvalues of a bidiagonal matrix, Amer. Math. Monthly, 111 (2004) 441
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LINKS
| L. E. Jeffery, Unit-primitive matrices
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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 (fvlamoen(AT)hotmail.com), Nov 02 2005
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MATHEMATICA
| LinearRecurrence[{1, 2, -1}, {1, 0, 1}, 60] (* From Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)
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CROSSREFS
| Cf. A006053, A052547.
Sequence in context: A032006 A167158 A074392 * A052547 A119245 A128731
Adjacent sequences: A096973 A096974 A096975 * A096977 A096978 A096979
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KEYWORD
| easy,nonn,changed
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Jul 16 2004
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