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A120757
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Expansion of x^2*(2+x)/(1-3*x-4*x^2-x^3).
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13
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0, 2, 7, 29, 117, 474, 1919, 7770, 31460, 127379, 515747, 2088217, 8455018, 34233669, 138609296, 561217582, 2272323599, 9200450421, 37251863241, 150829715006, 610697048403, 2472661868474, 10011603514040, 40536155064419
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OFFSET
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1,2
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COMMENTS
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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](n>=1).
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(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.
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 2 2012.
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REFERENCES
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P. Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, 70,1 (1997) 22-31.
R. Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5.
R. Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
R. Witula, Full Description of Ramanujan Cubic Polynomials, J. Integer Seq., 13 (2010), Article 10.5.7.
R. Witula, Ramanujan Cubic Polynomials of the Second Kind, J. Integer Seq., 13 (2010), Article 10.7.5.
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
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LINKS
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Table of n, a(n) for n=1..24.
Author?, Title?
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FORMULA
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a(n)=3a(n-1)+4a(n-2)+a(n-3) (follows from the minimal polynomial of the matrix M). See also the o.g.f. given in the name.
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EXAMPLE
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a(7)=1919 because M^7= [1919,3458,4312;3458,6231,7770;4312,7770,9689].
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MAPLE
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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);
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MATHEMATICA
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LinearRecurrence[{3, 4, 1}, {0, 2, 7}, 40] - Roman Witula, Aug 2 2012
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CROSSREFS
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C.f.A214683,A215076,A215100,A006053. - Roman Witula, Aug 2 2012.
Sequence in context: A155186 A203969 A199581 * A134169 A052961 A150662
Adjacent sequences: A120754 A120755 A120756 * A120758 A120759 A120760
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KEYWORD
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nonn
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AUTHOR
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Gary W. Adamson & Roger L. Bagula, Jul 01 2006
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EXTENSIONS
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Edited by N. J. A. Sloane, Dec 03 2006
New name, old name as comment; o.g.f.; reference.
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STATUS
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approved
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