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Search: id:A033304
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| A033304 |
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Expansion of (2+2*x-3*x^2)/(1-2*x-x^2+x^3). |
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+0
5
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| 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; listen)
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OFFSET
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0,1
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics I, p. 244, Eq. (36).
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.3.
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FORMULA
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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 (vladeta(AT)eunet.rs), Jun 27 2001
a(n) = trance 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)-th powers of the roots of the corresponding characteristic polynomial: -x^3 + 2x^2 + x -1 = 0. a(n) = A006356(n) + A006356(n-1) + 2*A006356(n-2)/ where A006356 = 1, 3, 6, 14, 31, 70, 157... E.g. a(3) = 26 = the trace of M^4. The characteristic polynomial of this matrix (see A066170) is -x^3 + 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 (qntmpkt(AT)yahoo.com), Feb 01 2004
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PROGRAM
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(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 */
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CROSSREFS
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Cf. A066170, A006356.
A096975(n)=a(-1-n). - Michael Somos Aug 03 2006.
Sequence in context: A079118 A034466 A007186 this_sequence A091622 A165821 A135048
Adjacent sequences: A033301 A033302 A033303 this_sequence A033305 A033306 A033307
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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