A096976 - OEIS

%I #41 Dec 15 2023 19:18:18

%S 1,0,1,0,2,1,5,5,14,19,42,66,131,221,417,728,1341,2380,4334,7753,

%T 14041,25213,45542,81927,147798,266110,479779,864201,1557649,2806272,

%U 5057369,9112264,16420730,29587889,53317085,96072133,173118414,311945595,562110290

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

%C 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.

%C 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

%C 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

%H Michael De Vlieger, <a href="/A096976/b096976.txt">Table of n, a(n) for n = 0..3914</a>

%H Robin Chapman and Nicholas C. Singer, <a href="http://www.jstor.org/stable/4145281">Eigenvalues of a bidiagonal matrix</a>, Amer. Math. Monthly, 111 (2004) 441.

%H Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, <a href="https://arxiv.org/abs/2210.12411">On a variant of Flory model</a>, arXiv:2210.12411 [math.CO], 2022.

%H L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>

%H Kai Wang, <a href="https://www.researchgate.net/publication/337943524_Fibonacci_Numbers_And_Trigonometric_Functions_Outline">Fibonacci Numbers And Trigonometric Functions Outline</a>, (2019).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1).

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

%F a(n) = 5a(n-2)-6a(n-4)+a(n-6). - _Floor van Lamoen_, Nov 02 2005

%F a(n) = A077998(-n) for all n in Z. - _Michael Somos_, Dec 12 2023

%e 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

%t LinearRecurrence[{1, 2, -1}, {1, 0, 1}, 60] (* _Vladimir Joseph Stephan Orlovsky_, Feb 13 2012 *)

%t a[ n_] := {1, 0, 0} . MatrixPower[{{1, 2, -1}, {1, 0, 0}, {0, 1, 0}}, n] . {1, 1, 3}; (* _Michael Somos_, Dec 12 2023 *)

%o (PARI) {a(n) = [1, 0, 0] * [1, 2, -1; 1, 0, 0; 0, 1, 0]^n * [1, 1, 3]~}; /* _Michael Somos_, Dec 12 2023 */

%Y Cf. A006053, A052547, A077998.

%K easy,nonn

%O 0,5

%A _Paul Barry_, Jul 16 2004