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 A033304 Expansion of (2 + 2*x - 3*x^2) / (1 - 2*x - x^2 + x^3). 19
 2, 6, 11, 26, 57, 129, 289, 650, 1460, 3281, 7372, 16565, 37221, 83635, 187926, 422266, 948823, 2131986, 4790529, 10764221, 24186985, 54347662, 122118088, 274396853, 616564132, 1385407029, 3112981337, 6994805571, 15717185450 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS From L. Edson Jeffery, Mar 22 2011: (Start) Let A be the unit-primitive matrix (see [Jeffery]) A=A_(7,2)= (0 0 1) (0 1 1) (1 1 1). Let B={b(n)} be this sequence shifted to the right one place and setting b(0)=3. Then B=(3,2,6,11,26,...) with generating function (3-4*x-x^2)/(1-2*x-x^2+x^3) and b(n)=Trace(A^n). (End) The following identity hold true (a(n)^2 - a(2n+2))/2 = A094648(n+1) = (-1)^(n+1)*A096975(n+1) - for the proof see Witula et al.'s papers - Roman Witula, Jul 25 2012 We note that the joined sequences (-1)^(n+1)*a(n) and A094648(n) form a two-sided sequence defined either by the recurrence formula x(n+3) + x(n+2) - 2x(n+1) - x(n) = 0, n in Z, x(0)=3, x(-1)=-2, x(1)=-1, or by the following trigonometric identities: x(n) = (c(1))^n + (c(2))^n + (c(4))^n = (c(1)c(2))^(-n) + (c(1)c(4))^(-n) + (c(2)c(4))^(-n) = (s(2)/s(1))^n + (s(4)/s(2))^n + (s(1)/s(4))^n, for n in Z, where c(j) := 2*cos(2Pi*j/7) and s(j) := sin(2*Pi*j/7) - for the proof see Witula's and Witula et al.'s papers. - Roman Witula, Jul 25 2012 We have 4*a(n+2) - a(n) = 7*A077998(n+2). - Roman Witula, Aug 13 2012 Two very intriguing identities of trigonometric nature hold: (-1)^n*(a(n)-a(n-1)) = c(1)*c(2)^(-n) + c(2)*c(4)^(-n) + c(4)*c(1)^(-n), and (-1)^(n+1)*(a(n-1)-a(n+1)) = c(1)*c(4)^(-n-1) + c(2)*c(1)^(-n-1) + c(4)*c(2)^(-n-1), where a(-1):=3 and c(j) is defined as above. For the proof see Remark 6 in the first Witula's paper. - Roman Witula, Aug 14 2012 With respect to the form of the trigonometric formulas describing a(n), we call this sequence the Berndt-type sequence number 20 for the argument 2Pi/7. The A-numbers of other Berndt-type sequences numbers are given in below. - Roman Witula, Sep 30 2012 REFERENCES R. P. Stanley, Enumerative Combinatorics I, p. 244, Eq. (36). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (2,1,-1) Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, John Pryce, On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials, arXiv:1502.03085 [math.NT], 2015 (see p. 40). L. E. Jeffery, Unit-primitive matrices Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5. Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6 Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3. FORMULA a(-1-n) = A096975(n). a(n) = (1-2*cos(1/7*Pi))^(n+1)+(1+2*cos(2/7*Pi))^(n+1)+(1-2*cos(3/7*Pi))^(n+1). - Vladeta Jovovic, Jun 27 2001 a(n) = trace of (n+1)-th power of the 3 X 3 matrix (in the example of A066170): [1 1 1 / 1 1 0 / 1 0 0]. Alternatively, the sum of the (n+1)st powers of the roots of the corresponding characteristic polynomial: x^3 - 2*x^2 - x + 1 = 0. a(n) = A006356(n) + A006356(n-1) + 2*A006356(n-2). E.g., a(3) = 26 = the trace of M^4. The characteristic polynomial of this matrix (see A066170) is x^3 - 2*x^2 - x + 1 and the roots are 2.24697960372..., -.8019377358...and .55495813208...= a, b, c. Then Sum(a^4 + b^4 + c^4) = 26. - Gary W. Adamson, Feb 01 2004 (-1)^(n+1)*a(n) = (c(1))^(-n-1) + (c(2))^(-n-1) + (c(3))^(-n-1) = (c(1)c(2))^(n+1) + (c(1)c(4))^(n+1) + (c(2)c(4))^(n+1) = (s(1)/s(2))^(n+1) + (s(2)/s(4))^(n+1) + (s(4)/s(1))^(n+1), where c(j) := 2*cos(2*Pi*j/7) and s(j) := sin(2*Pi*j/7) - for the proof see Witula's and Witula et al.'s papers. - Roman Witula, Jul 25 2012 a(n) = 3*A077998(n+1)-A006054(n+2)-A006054(n+1). - Roman Witula, Aug 13 2012 a(n)*(-1)^(n+1) = (A094648(n+1)^2 - A094648(2*(n+1)))/2. - Roman Witula, Sep 30 2012 MATHEMATICA CoefficientList[Series[(2+2x-3x^2)/(1-2x-x^2+x^3), {x, 0, 50}], x]  (* Harvey P. Dale, Mar 14 2011 *) LinearRecurrence[{2, 1, -1}, {2, 6, 11}, 29] (* Jean-François Alcover, Sep 27 2017 *) PROG (PARI) {a(n)=if(n<0, n=-n; polsym(x^3-x^2-2*x+1, n-1)[n], n+=2; polsym(1-x-2*x^2+x^3, n-1)[n])} /* Michael Somos, Aug 03 2006 */ (PARI) x='x+O('x^99); Vec((2+2*x-3*x^2)/(1-2*x-x^2+x^3)) \\ Altug Alkan, Apr 19 2018 (MAGMA) I:=[2, 6, 11]; [n le 3 select I[n] else 2*Self(n-1) +Self(n-2) - Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 19 2018 CROSSREFS Cf. A066170, A006356, A096975. Cf. A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215575, A215694, A215695, A108716, A215794, A215828, A215817, A215877, A094429, A094430, A217274, A094648. Sequence in context: A034466 A191307 A007186 * A091622 A191315 A184884 Adjacent sequences:  A033301 A033302 A033303 * A033305 A033306 A033307 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified December 15 00:30 EST 2019. Contains 329988 sequences. (Running on oeis4.)