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.
Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, On a variant of Flory model, arXiv:2210.12411 [math.CO], 2022.
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
a(n) = A077998(-n) for all n in Z. - Michael Somos, Dec 12 2023
EXAMPLE
G.f. = 1 + x^2 + 2*x^4 + x^5 + 5*x^6 + 5*x^7 + 14*x^8 + 19*x^9 + ... - Michael Somos, Dec 12 2023
MATHEMATICA
LinearRecurrence[{1, 2, -1}, {1, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)
a[ n_] := {1, 0, 0} . MatrixPower[{{1, 2, -1}, {1, 0, 0}, {0, 1, 0}}, n] . {1, 1, 3}; (* Michael Somos, Dec 12 2023 *)
PROG
(PARI) {a(n) = [1, 0, 0] * [1, 2, -1; 1, 0, 0; 0, 1, 0]^n * [1, 1, 3]~}; /* Michael Somos, Dec 12 2023 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jul 16 2004
STATUS
approved