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%I #46 Nov 26 2022 02:44:06
%S 0,2,7,29,117,474,1919,7770,31460,127379,515747,2088217,8455018,
%T 34233669,138609296,561217582,2272323599,9200450421,37251863241,
%U 150829715006,610697048403,2472661868474,10011603514040,40536155064419
%N Expansion of x^2*(2+x)/(1-3*x-4*x^2-x^3).
%C The (1,1)-entry of the matrix M^n, where M is the 3 X 3 matrix [0,1,1; 1,1,2; 1,2,2].
%C a(n)/a(n-1) tends to 4.0489173...an eigenvalue of M and a root to the characteristic polynomial x^3 - 3x^2 - 4x - 1.
%C C(n):=a(n), with a(0):=1 (hence the o.g.f. for C(n) is (1-3*x-2*x^2)/(1-3*x-4*x^2-x^3)), appears in the following formula for the nonnegative powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7) = rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis <1,rho,sigma> is used in an extension of the rational field. (rho*sigma)^n = C(n) + B(n)*rho + A(n)*sigma,n>=0, with B(n)= |A122600(n-1)|, B(0)=0, and A(n)= A181879(n). For the nonpositive powers see A085810(n)*(-1)^n, A181880(n-2)*(-1)^n and A116423(n+1)*(-1)^(n+1), respectively. See also a comment under A052547.
%C We have a(n)=cs(3n+1), where the sequence cs(n) and its two conjugate sequences as(n) and bs(n) are defined in the comments to the sequence A214683 (see also A215076, A215100, A006053). We call the sequence a(n) the Ramanujan-type sequence number 5 for the argument 2Pi/7. Since as(3n+1)=bs(3n+1)=0, we obtain the following relation: 49^(1/3)*a(n) = (c(1)/c(4))^(n + 1/3) + (c(4)/c(2))^(n + 1/3) + (c(2)/c(1))^(n + 1/3), where c(j) := Cos(2Pi/7) (for more details and proofs see Witula et al.'s papers). - _Roman Witula_, Aug 02 2012
%D R. Witula, E. Hetmaniok and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.
%H G. C. Greubel, <a href="/A120757/b120757.txt">Table of n, a(n) for n = 1..1000</a>
%H P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31, <a href="http://www.ams.org/mathscinet-getitem?mr=1439165">MR 1439165</a>
%H Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Witula/witula17.html">Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
%H Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Witula/witula30.html">Full Description of Ramanujan Cubic Polynomials</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
%H Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Witula2/witula40r.html">Ramanujan Cubic Polynomials of the Second Kind</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
%H Roman Witula, <a href="https://doi.org/10.1515/dema-2013-0418">Ramanujan Type Trigonometric Formulae</a>, Demonstratio Math. 45 (2012) 779-796.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,4,1).
%F a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3) (follows from the minimal polynomial of the matrix M). See also the o.g.f. given in the name.
%e a(7)=1919 because M^7= [1919,3458,4312;3458,6231,7770;4312,7770,9689].
%p with(linalg): M[1]:=matrix(3,3,[0,1,1,1,1,2,1,2,2]): for n from 2 to 25 do M[n]:=multiply(M[1],M[n-1]) od: seq(M[n][1,1],n=1..25);
%t LinearRecurrence[{3,4,1},{0,2,7},40] (* _Roman Witula_, Aug 02 2012 *)
%o (PARI) a(n)=([0,1,0; 0,0,1; 1,4,3]^(n-1)*[0;2;7])[1,1] \\ _Charles R Greathouse IV_, Jun 22 2016
%o (Magma) a:=[0,2,7]; [ n le 3 select a[n] else 3*Self(n-1) + 4*Self(n-2) + Self(n-3): n in [1..25]]; // _Marius A. Burtea_, Oct 03 2019
%o (SageMath)
%o @CachedFunction
%o def a(n): # a = A120757
%o if (n<3): return (0,2,7)[n]
%o else: return 3*a(n-1) + 4*a(n-2) + a(n-3)
%o [a(n) for n in range(40)] # _G. C. Greubel_, Nov 25 2022
%Y Cf. A006053, A214683, A215076, A215100.
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_ & _Roger L. Bagula_, Jul 01 2006
%E Edited by _N. J. A. Sloane_, Dec 03 2006
%E New name, old name as comment; o.g.f.; reference.
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