cp's OEIS Frontend

A033304 Expansion of (2 + 2*x - 3*x^2) / (1 - 2*x - x^2 + x^3).

%I A033304 #87 Sep 18 2024 08:36:52
%S A033304 2,6,11,26,57,129,289,650,1460,3281,7372,16565,37221,83635,187926,
%T A033304 422266,948823,2131986,4790529,10764221,24186985,54347662,122118088,
%U A033304 274396853,616564132,1385407029,3112981337,6994805571,15717185450
%N A033304 Expansion of (2 + 2*x - 3*x^2) / (1 - 2*x - x^2 + x^3).
%C A033304 From _L. Edson Jeffery_, Mar 22 2011: (Start)
%C A033304 Let A be the unit-primitive matrix (see [Jeffery])
%C A033304 A=A_(7,2)=
%C A033304 (0 0 1)
%C A033304 (0 1 1)
%C A033304 (1 1 1).
%C A033304 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)
%C A033304 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
%C A033304 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
%C A033304 We have 4*a(n+2) - a(n) = 7*A077998(n+2). - _Roman Witula_, Aug 13 2012
%C A033304 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
%C A033304 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
%D A033304 R. P. Stanley, Enumerative Combinatorics I, p. 244, Eq. (36).
%H A033304 G. C. Greubel, <a href="/A033304/b033304.txt">Table of n, a(n) for n = 0..1000</a>
%H A033304 Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, John Pryce, <a href="http://arxiv.org/abs/1502.03085">On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials</a>, arXiv:1502.03085 [math.NT], 2015 (see p. 40).
%H A033304 L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H A033304 Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Witula/witula17.html">Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7</a>, J. Integer Seq., 12 (2009), Article 09.8.5.
%H A033304 Roman Witula and Damian Slota, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota/witula13.html">New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
%H A033304 Roman Witula, Damian Slota and Adam Warzynski, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Slota/slota57.html">Quasi-Fibonacci Numbers of the Seventh Order</a>, J. Integer Seq., 9 (2006), Article 06.4.3.
%H A033304 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-1).
%F A033304 a(-1-n) = A096975(n).
%F A033304 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
%F A033304 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..., -0.8019377358... and 0.55495813208... = a, b, c. Then Sum(a^4 + b^4 + c^4) = 26. - _Gary W. Adamson_, Feb 01 2004
%F A033304 (-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
%F A033304 a(n) = 3*A077998(n+1) - A006054(n+2) - A006054(n+1). - _Roman Witula_, Aug 13 2012
%F A033304 a(n)*(-1)^(n+1) = (A094648(n+1)^2 - A094648(2*(n+1)))/2. - _Roman Witula_, Sep 30 2012
%t A033304 CoefficientList[Series[(2+2x-3x^2)/(1-2x-x^2+x^3),{x,0,50}], x]  (* _Harvey P. Dale_, Mar 14 2011 *)
%t A033304 LinearRecurrence[{2, 1, -1}, {2, 6, 11}, 29] (* _Jean-François Alcover_, Sep 27 2017 *)
%o A033304 (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 */
%o A033304 (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
%o A033304 (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
%Y A033304 Cf. A066170, A006356, A096975.
%Y A033304 Cf. A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215575, A215694, A215695, A108716, A215794, A215828, A215817, A215877, A094429, A094430, A217274, A094648.
%K A033304 nonn,easy
%O A033304 0,1
%A A033304 _N. J. A. Sloane_