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 A096975 Trace sequence of a path graph plus loop. 5
 3, 1, 5, 4, 13, 16, 38, 57, 117, 193, 370, 639, 1186, 2094, 3827, 6829, 12389, 22220, 40169, 72220, 130338, 234609, 423065, 761945, 1373466, 2474291, 4459278, 8034394, 14478659, 26088169, 47011093, 84708772, 152642789, 275049240 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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)). From L. Edson Jeffery, Mar 22 2011: (Start) Let A be the unit-primitive matrix (see [Jeffery]) A = A_(7,1) = (0 1 0) (1 0 1) (0 1 1). Then a(n) = Trace(A^n). (End) LINKS Michael De Vlieger, Table of n, a(n) for n = 0..3910 A. Akbary, Q. Wang, A generalized Lucas sequence and permutations binomials, Proc. Am. Math. Soc. 134 (2006) 15-22, sequence a(n) with l=7. Robin Chapman and Nicholas C. Singer, Eigenvalues of a bidiagonal matrix, Amer. Math. Monthly, 111 (2004), p. 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 matrix Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019. Q. Wang, On generalized Lucas sequences, Contemp. Math. 531 (2010) 127-141, Table 1 (k=3). Index entries for linear recurrences with constant coefficients, signature (1,2,-1). FORMULA G.f.: (3-2*x-2*x^2)/(1-x-2*x^2+x^3); a(n) = a(n-1) + 2*a(n-2) - a(n-3); 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. a(n) = 2^n*((cos(Pi/7))^n+(cos(3*Pi/7))^n+(cos(5*Pi/7))^n). - Vladimir Shevelev, Aug 25 2010 MATHEMATICA 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 *) PROG (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 */ (PARI) a(n)=trace([0, 1, 0; 1, 0, 1; 0, 1, 1]^n); /* Joerg Arndt, Apr 30 2011 */ CROSSREFS Cf. A006053, A052547, A096976. A033304(n) = a(-1-n). - Michael Somos, Aug 03 2006 Sequence in context: A096374 A007085 A094648 * A145174 A351965 A228785 Adjacent sequences: A096972 A096973 A096974 * A096976 A096977 A096978 KEYWORD easy,nonn AUTHOR Paul Barry, Jul 16 2004 STATUS approved

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Last modified September 22 07:52 EDT 2023. Contains 365519 sequences. (Running on oeis4.)