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%I #53 Oct 26 2022 19:22:45
%S 3,1,5,4,13,16,38,57,117,193,370,639,1186,2094,3827,6829,12389,22220,
%T 40169,72220,130338,234609,423065,761945,1373466,2474291,4459278,
%U 8034394,14478659,26088169,47011093,84708772,152642789,275049240
%N Trace sequence of a path graph plus loop.
%C Let A be the adjacency matrix of the graph P_3 with a loop added at the end. Then a(n) = trace(A^n). A is a 'reverse Jordan matrix' [0,0,1;0,1,1;1,1,0]. a(n) = abs(A094648(n)).
%C From _L. Edson Jeffery_, Mar 22 2011: (Start)
%C Let A be the unit-primitive matrix (see [Jeffery])
%C A = A_(7,1) =
%C (0 1 0)
%C (1 0 1)
%C (0 1 1).
%C Then a(n) = Trace(A^n). (End)
%H Michael De Vlieger, <a href="/A096975/b096975.txt">Table of n, a(n) for n = 0..3910</a>
%H A. Akbary, Q. Wang, <a href="http://dx.doi.org/10.1090/S0002-9939-05-08220-1">A generalized Lucas sequence and permutations binomials</a>, Proc. Am. Math. Soc. 134 (2006) 15-22, sequence a(n) with l=7.
%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), p. 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 matrix</a>
%H Genki Shibukawa, <a href="https://arxiv.org/abs/1907.00334">New identities for some symmetric polynomials and their applications</a>, arXiv:1907.00334 [math.CA], 2019.
%H Q. Wang, <a href="https://www.semanticscholar.org/paper/On-generalized-Lucas-sequences-Wang-Akbari/7e33b3b79703dc6790fca133e8c92cc0cafcfe4a">On generalized Lucas sequences</a>, Contemp. Math. 531 (2010) 127-141, Table 1 (k=3).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1).
%F G.f.: (3-2*x-2*x^2)/(1-x-2*x^2+x^3);
%F a(n) = a(n-1) + 2*a(n-2) - a(n-3);
%F a(n) = (2*sqrt(7)*sin(atan(sqrt(3)/9)/3)/3+1/3)^n + (1/3-2*sqrt(7)*sin(atan(sqrt(3)/9)/3+Pi/3)/3)^n + (2*sqrt(7)*cos(acot(-sqrt(3)/9)/3)/3+1/3)^n.
%F a(n) = 2^n*((cos(Pi/7))^n+(cos(3*Pi/7))^n+(cos(5*Pi/7))^n). - _Vladimir Shevelev_, Aug 25 2010
%t CoefficientList[Series[(3 - 2 x - 2 x^2)/(1 - x - 2 x^2 + x^3), {x, 0, 33}], x] (* _Michael De Vlieger_, Aug 21 2019 *)
%o (PARI) {a(n)=if(n>=0, n+=1; polsym(x^3-x^2-2*x+1,n-1)[n], n=1-n; polsym(1-x-2*x^2+x^3,n-1)[n])} /* _Michael Somos_, Aug 03 2006 */
%o (PARI) a(n)=trace([0,1,0;1,0,1;0,1,1]^n); /* _Joerg Arndt_, Apr 30 2011 */
%Y Cf. A006053, A052547, A096976.
%Y A033304(n) = a(-1-n). - _Michael Somos_, Aug 03 2006
%K easy,nonn
%O 0,1
%A _Paul Barry_, Jul 16 2004
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